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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈ ▼❡str❛❞♦ ♣r♦❢✐ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ♠❛t❡♠át✐❝❛ ✲ P❘❖❋▼❆❚ ❉✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦ ❋❧❛✈✐♦ ❋❡r♥❛♥❞♦ ❞❛ ❙✐❧✈❛ ❆r❜❡❧♦s ❙❛♥t♦ ❆♥❞ré ✲ ❙P ✷✵✶✹✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈ ❈❡♥tr♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❈♦♠♣✉t❛çã♦ ❡ ❈♦❣♥✐çã♦ ❆r❜❡❧♦s ❋❧❛✈✐♦ ❋❡r♥❛♥❞♦ ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧✐③❛♥t❡ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❆❇❈✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❙❛♥t♦ ❆♥❞ré ✲ ❙P ❆❣♦st♦ ❞❡ ✷✵✶✹✳ ❆r❜❡❧♦s ❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à r❡❞❛çã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡✈✐❞❛♠❡♥t❡ ❝♦rr✐✲ ❣✐❞❛ ❡ ❞❡❢❡♥❞✐❞❛ ♣♦r ❋❧❛✈✐♦ ❋❡r♥❛♥❞♦ ❞❛ ❙✐❧✈❛ ❡ ❛♣r♦✈❛❞❛ ♣❡❧❛ ❝♦♠✐ssã♦ ❥✉❧❣❛❞♦r❛✳ ❙❛♥t♦ ❆♥❞ré✱ ✷✻ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✶✹✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r ❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿ ✶✳ Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛ ✭❖r✐❡♥t❛❞♦r✮ ✲ ❯❋❆❇❈ ✷✳ Pr♦❢✳ ❉r✳ ❙✐♥✉❡ ❉❛②❛♥ ❇❛r❜❡r♦ ▲✉❞♦✈✐❝✐ ✲ ❯❋❆❇❈ ✸✳ Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ▲②♠❜❡r♦♣♦✉❧♦s ✲ ❯❙P ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❥✉♥t♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❆❇❈✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥✲ çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ à ♠✐♥❤❛ ❡s♣♦s❛✱ ♠❡✉s ♣❛✐s✱ ❡ ❛♠✐❣♦s❀ ❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ ♠❡ ❛♣♦✐❛r❛♠ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛✳ ❆❣r❛❞❡❝✐♠❡♥t♦s Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t✉❞♦✳ ❆♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✱ à ❈❆P❊❙ ♣❡❧♦ ❛✉①í❧✐♦ ❝♦♥❝❡❞✐❞♦✱ à ❯❋❆❇❈ ❡ s❡✉s ♣r♦❢❡ss♦r❡s✱ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár❝✐♦ ❋❛❜✐❛♥♦ ❞❛ ❙✐❧✈❛✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❛♦ ❛❝❡✐t❛r ♦ ♣❡❞✐❞♦ ❞❡ s❡r ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♣♦r ❛❝r❡❞✐t❛r ❡♠ ♠❡✉ ♣♦tê♥❝✐❛❧✱ ♣♦r s❡r ❝♦♠♣❡t❡♥t❡ ❡ ❡①❝❡❧❡♥t❡ ❡♠ s✉❛ ♣r♦✜ssã♦✱ ❛♦s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❞❡ ♠❡str❛❞♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ à ♠✐♥❤❛ ❡s♣♦s❛ ❘❛q✉❡❧ q✉❡ q✉❡ ♠❡ ✐❝❡♥t✐✈♦✉ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦✱ ❛❜r✐♥❞♦ ♠ã♦ ❞❡ ♣❛ss❡✐♦s ❡ ❧❛s❡r✱ ♥ã♦ ♠❡❞✐♥❞♦ ❡s❢♦rç♦ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r ❛q✉✐ ❡ ♠❡✉ ❛♠✐❣♦ ▲✉❝✐❛♥♦ q✉❡ t❛♥t♦ ♠❡ ❛❥✉❞♦✉ ❝♦♠ ♦s ❡st✉❞♦s✱ ✈✐❛❥❡♥s ❡ s✉❛s ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦✳ ❆ t♦❞♦s ✈♦❝ês✱ s✐♥❝❡r❛ ❣r❛t✐❞ã♦✳ ✈✐✐ ❘❡s✉♠♦ ■♥s♣✐r❛❞♦ ♥♦ ❛rt✐❣♦ ❞❡ ❍❛r♦❧❞ P✳ ❇♦❛s ❬✷❪✱ ♥❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ♦s ❆r❜❡❧♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❆♥❛❧✐s❛♠♦s ❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❡ ❛♣❧✐❝❛♠♦s ❡ss❛ té❝♥✐❝❛ ♥❛ ❝♦♥str✉çã♦ ❞❛ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s ❡ ❞♦ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ P❛❧❛✈r❛s✲❈❤❛✈❡ ❆r❜❡❧♦s✱ ❆r❜❡❧♦s ●ê♠❡♦s✱ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s✱ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✱ ●❡♦♠❡tr✐❛ ■♥✈❡rs✐✈❛ ✈✐✐✐ ❆❜str❛❝t ❇❛s❡❞ ♦♥ t❤❡ ✇♦r❦ ♦❢ ❍❛r♦❧❞ P✳ ❇♦❛s ❬✷❪✱ ✐♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t❤❡ ❛r❜❡❧♦s ❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s✳ ✇❡ ❛♥❛❧②s❡ t❤❡ ✐♥✈❡rs✐♦♥ ❛❜♦✉t ❛ ❝✐r❝❧❡ ❛♥❞ ❛♣♣❧② t❤✐s t❡❝❤♥✐q✉❡ t♦ t❤❡ ❝♦♥s✲ tr✉❝t✐♦♥ ♦❢ P❛♣♣✉s ❈❤❛✐♥ ❛♥❞ ❇❛♥❦♦✛✬s ❝✐r❝❧❡✳ ❑❡②✇♦r❞s ❆r❜❡❧♦s✱ ❚✇✐♥ ❛r❜❡❧♦s✱ P❛♣♣✉s ❈❤❛✐♥✱ ❇❛♥❦♦✛✬s ❝✐r❝❧❡✱ ✐♥✈❡rs✐✈❡ ❣❡♦♠❡tr②✳ ❙✉♠ár✐♦ ✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✺ ✷ ●❡♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛ ✷✼ ✷✳✶ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ■♥✈❡rsã♦ ❞❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ■♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❞❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸ ❆r❜❡❧♦s ✸✾ ✸✳✶ ❆r❜❡❧♦s ❣ê♠❡♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✸ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹ ❆♣❧✐❝❛çõ❡s ✺✼ ✹✳✶ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✐① ① ❙❯▼➪❘■❖ ▲✐st❛ ❞❡ ❋✐❣✉r❛s ✶✳✶ ❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k = 2. ✶✳✷ ❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k = −0, 6. ✳ ✶✳✸ ❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ❡①t❡r♥❛s ❞❡ Γ ❡ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✹ ❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ✐♥t❡r♥❛s ❞❡ Γ ❡ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✺ ❈ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✻ ❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ r❡t❛ ❡♠ r❡t❛ ❞❡ r❛③ã♦ ❦❂✶✳✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✼ ❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ ❝ír❝✉❧♦ ❡♠ ❝ír❝✉❧♦ ❞❡ r❛③ã♦ ❦❂✶✳✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✽ ❍♦♠♦t❡t✐❛ ❞❡ r❛③ã♦ ❦❂✶✳✾ ✶✾ ✶✳✾ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ P ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✶✵ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ P ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✶✸ ❈ír❝✉❧♦ Γ ♦rt♦❣♦♥❛❧ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ Ω ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✹ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✺ ❚r✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦ ✶✳✶✻ ❊❧✐♣s❡ ❞❡ ❢♦❝♦s ✶✳✶✼ ❚r✐â♥❣✉❧♦ ❆❇❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✶ ❙❡❣♠❡♥t♦ P❚ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ ✶✳✶✷ ❘❡t❛ ❡ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✻ ✷✷ ✷✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ F1 ❡ F2 ✳ ✷✳✶ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ O ♥ã♦ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✹ ❖ ♣♦♥t♦ P é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✺ ❖ ♣♦♥t♦ P é ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ①✐ ✷✼ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✶✷ ✷✳✻ ✷✳✼ ✷✳✽ ✷✳✾ ✷✳✶✵ ✷✳✶✶ ✷✳✶✷ ✷✳✶✸ ✷✳✶✹ ✷✳✶✺ ✷✳✶✻ ✷✳✶✼ ✷✳✶✽ ❖ ♣♦♥t♦ P ♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ■♥✈❡rsã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳ ✳ ✳ ✳ ❆ r❡t❛ s é ❡①t❡r♥❛ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆ r❡t❛ s é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆ r❡t❛ s é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ■♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✶ ✸✷ ✸✷ ✸✷ ✸✸ ✸✹ ✸✹ ✸✺ ✸✺ ✸✺ ✸✻ ✸✼ ✸✳✶ ✸✳✷ ✸✳✸ ✸✳✹ ✸✳✺ ✸✳✻ ✸✳✼ ✸✳✽ ✸✳✾ ✸✳✶✵ ✸✳✶✶ ✸✳✶✷ ✸✳✶✸ ✸✳✶✹ ❆r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆r❜❡❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♦r❞❡♥❛❞❛s ❞♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥str✉çã♦ ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ✉s❛♥❞♦ ✐♥✈❡rsã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ❛té ❛ r❡t❛ s✉♣♦rt❡✳ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✵ ✹✹ ✹✹ ✹✼ ✹✼ ✹✽ ✹✽ ✺✵ ✺✵ ✺✶ ✺✷ ✺✸ ✺✹ ✹✳✶ ✹✳✷ ✹✳✸ ✹✳✹ ✹✳✺ ❱✐s✉❛❧✐③❛çã♦ ❞❛ ♣r✐♠❡✐r❛ ❝♦♥str✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱✐s✉❛❧✐③❛çã♦ ❞❛ s❡❣✉♥❞❛ ❝♦♥str✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ❱✐s✉❛❧✐③❛çã♦ ❞❡ ❝♦♠♦ ✜❝❛r✐❛ ❞❡♣♦✐s ❞❡ r❡s♦❧✈✐❞♦✳ ■♥✈❡rt❡♥❞♦ r ❡♠ r❡❧❛çã♦ ❛ C ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙♦❧✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✻✵ ✻✶ ✻✷ ✻✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ■♥tr♦❞✉çã♦ ❖s ❛r❜❡❧♦s ❢♦r❛♠ ❡st✉❞❛❞♦s ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ❆rq✉✐♠❡❞❡s ❡♠ s❡✉ ▲✐✈r♦ ❞❡ ▲❡♠❛s✱ P❛♣♣✉s t❛♠❜é♠ ❡st✉❞♦✉ ♦s ❛r❜❡❧♦s ♥♦ ❧✐✈r♦ ■❱ ❞❡ s✉❛ ❝♦❧❡çã♦✳ ❊st❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ✹ ❝❛♣ít✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✶ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s ❝♦♠♦✿ ❤♦♠♦t❡t✐❛✱ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦✱ r❡t❛s ❡ ❝ír❝✉❧♦s ♦rt♦✲ ❣♦♥❛✐s✱ ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✱ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦✱ ❡❧✐♣s❡ ❡ ❛ ❢ór♠✉❧❛ ❞❡ ❍❡r♦♥✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ❛❜♦r❞❛r❡♠♦s ❛ ❣❡♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛✳ ❙❡rá ❞❡✜♥✐❞♦ ♦ q✉❡ é ✐♥✈❡rsã♦ ❡ ♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ ✐♥✈❡rt❡♠♦s ✉♠❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❡ ✉♠ ❝ír❝✉❧♦ ❡♠ r❡❧❛çã♦ ❛ ♦✉tr♦✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ s❡rá ❢❡✐t♦ ♦ ❡st✉❞♦ s♦❜r❡ ❛r❜❡❧♦s✱ ❛ss✐♠ ❝♦♠♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐✲ ❡❞❛❞❡s ❡ ❛✐♥❞❛ ✉s❛r❡♠♦s ❛ ✐♥✈❡rsã♦ ♣❛r❛ ❢❛❧❛r ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ❡ ❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ◆♦ ❝❛♣ít✉❧♦ ✹✱ s❡rã♦ ❛♣❧✐❝❛❞♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s ♣❛r❛ ♣r♦✲ ❞✉③✐r ❡①❡♠♣❧♦s ❞❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ♣♦❞❡♠ s❡r ✉t✐✲ ❧✐③❛❞♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ P❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❛s ❛t✐✈✐❞❛❞❡s✱ s✉❣❡r✐♠♦s ♦ ✉s♦ ❞❡ ❛❧❣✉♠ s♦❢t✇❛r❡ ❞❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✱ ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ q✉❡ ❢♦✐ ✉t✐❧✐③❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦✱ ♠❛s ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ♣♦❞❡♠ s❡r r❡❛❧✐③❛❞❛s ✉s❛♥❞♦ ❛♣❡♥❛s ré❣✉❛ ❡ ❝♦♠♣❛ss♦✳ ❊st❛ é ✉♠❛ ❜♦❛ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦s ❛❧✉♥♦s ❡ ❤❛❜✐❧✐❞❛❞❡ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s q✉❡ ✉s❛♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❡ ❛❧❣é❜r✐❝♦✳ ✶✸ ✶✹ ▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ❈❛♣ít✉❧♦ ✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✷ tr❛t❛r❡♠♦s ❞❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✳ P❛r❛ ✐st♦✱ ♣r❡❝✐s❛r❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❤♦♠♦t❡t✐❛ ❡ ❞❡ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✱ ❛❧é♠ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝ír❝✉❧♦s ♦rt♦❣♦♥❛✐s✱ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r✳ P♦r ❝ír❝✉❧♦ q✉❡r❡♠♦s ❞✐③❡r ♦ ❧✉❣❛r ❣❡♦♠étr✐❝♦ ❞♦s ♣♦♥t♦s ♥♦ ♣❧❛♥♦ q✉❡ ❡q✉✐❞✐st❛♠ ❞❡ ✉♠ ❞❛❞♦ ♣♦♥t♦✳ ❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ F ✉♠❛ r❡❣✐ã♦ ❞♦ ♣❧❛♥♦✱ O ✉♠ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❡ k ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲♥✉❧♦✳ ❆ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k é ❛ tr❛♥s❢♦r♠❛çã♦ ❣❡♦♠étr✐❝❛ q✉❡ ❛ss♦❝✐❛ ❛ −→ ❝❛❞❛ ♣♦♥t♦ P ❞❡ F ♦ ♣♦♥t♦ P ′ s♦❜r❡ ❛ s❡♠✐rr❡t❛ OP ✱ ❞❡ ♦r✐❣❡♠ O✱ t❛❧ q✉❡ OP ′ = k · OP ✳ ❖❜s❡r✈❛çã♦ ✶✳✶✳ ❙❡ k > 0✱ ❛ ❤♦♠♦t❡t✐❛ ❞❡ r❛③ã♦ k é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❤♦♠♦t❡t✐❛ ❞✐r❡t❛✱ ❡ s❡ k < 0✱ ❝♦♠♦ ❤♦♠♦t❡t✐❛ ✐♥✈❡rs❛✱ q✉❡ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ❛ ❝♦♠♣♦s✐çã♦ ❡♥tr❡ ❛ ❤♦♠♦t❡t✐❛ ❞✐r❡t❛ ❞❡ r❛③ã♦ −k > 0 ❝♦♠ ❛ r❡✢❡①ã♦ ❡♠ r❡❧❛çã♦ ❛♦ ♣♦♥t♦ O✳ ❋✐❣✉r❛ ✶✳✶✿ ❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k = 2. ✶✺ ✶✻ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ❋✐❣✉r❛ ✶✳✷✿ ❍♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k = −0, 6. ❉♦✐s ❝ír❝✉❧♦s sã♦ s❡♠♣r❡ ❤♦♠♦tét✐❝♦s✶ ✳ ◆❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ❡❧❡s ❛❞♠✐t❡♠ ❞✉❛s ❤♦♠♦t❡t✐❛s✱ ✉♠❛ ❞✐r❡t❛ ❡ ✉♠❛ ✐♥✈❡rs❛✳ ◆♦ ❝❛s♦ ❞❡ ❝ír❝✉❧♦s ❞✐s❥✉♥t♦s✱ ♦s ❝❡♥tr♦s ❞❡ ❤♦♠♦t❡t✐❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ sã♦ ❛s ✐♥t❡rs❡❝çõ❡s ❞❛s t❛♥❣❡♥t❡s ❝♦♠✉♥s ✐♥t❡r♥❛s ✷ ✭✐♥✈❡rs❛✮ ❡ ❞❛s t❛♥❣❡♥t❡s ❝♦♠✉♥s ❡①t❡r♥❛s ✸ ✭❞✐r❡t❛✮✳ ❊st❡s r❡s✉❧t❛❞♦s ❡stã♦ ♣r♦✈❛❞♦s ♥❛s ❞✉❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s ❡ ✐❧✉str❛❞♦s ♥❛s ✜❣✉r❛s ✭✶✳✸✮✱ ✭✶✳✹✮ ❡ ✭✶✳✺✮✳ ❖❜s❡r✈❛çã♦ ✶✳✷✳ ❙❡❥❛♠ Γ ❡ Ω ❞♦✐s ❝ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳ ❆s t❛♥❣❡♥t❡s ❝♦♠✉♥s ❡①t❡r♥❛s ❝r✉③❛♠✲s❡ ❡♠ O q✉❡ é ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛✳ Pr♦♣♦s✐çã♦ ✶✳✶✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ {O} = t1 ∩ t2 ✱ ✜❣✉r❛ ✭✶✳✸✮✱ ♦s ♣♦♥t♦s A ❡ B ♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s Γ ❡ Ω r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ X ❡ Y ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞❡ t1 ❝♦♠ ♦s ❝ír❝✉❧♦s Γ ❡ Ω ❝♦♠♦ ✐♥❞✐❝❛❞♦ ♥❛ ✜❣✉r❛ ✭✶✳✺✮✳ P❡❧♦ ❝❛s♦ ❞❡ s❡♠❡❧❤❛♥ç❛ AA t❡♠♦s q✉❡ △OAX ∼ △OBY ✳ ▲♦❣♦✱ BY OB = = k ⇒ BY = k · AX. OA AX ❈♦♠♦ O é ú♥✐❝♦✱ ❡♥tã♦ O é ♦ ❝❡♥tr♦ ❞❛ ❤♦♠♦t❡t✐❛✳ Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡❥❛♠ Γ ❡ Ω ❝ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳ ❆s t❛♥❣❡♥t❡s ❝♦♠✉♥s ✐♥t❡r♥❛s ❝r✉③❛♠✲s❡ ❡♠ O1 q✉❡ é ♦ ❝❡♥tr♦ ❞❡ ❤♦♠♦t❡t✐❛✳ ✶ ❉♦✐s ✷ P❛r❛ ❝ír❝✉❧♦s sã♦ ❤♦♠♦tét✐❝♦s s❡ ❡①✐st✐r ✉♠❛ ❤♦♠♦t❡t✐❛ q✉❡ ❛♣❧✐❝❛ ✉♠ s♦❜r❡ ♦✉tr♦✳ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s t❛♥❣❡♥t❡s✱ ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❛♦s ❞♦✐s ❝ír❝✉❧♦s ❡stã♦ ❡♠ s❡♠✐♣❧❛♥♦s ♦♣♦st♦s ❡♠ r❡❧❛çã♦ à r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s✳ ✸ P❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❞✉❛s t❛♥❣❡♥t❡s✱ ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❛♦s ❞♦✐s ❝ír❝✉❧♦s ❡stã♦ ♥♦ ♠❡s♠♦ s❡♠✐♣❧❛♥♦ ❡♠ r❡❧❛çã♦ à r❡t❛ ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s✳ ✶✼ ❋✐❣✉r❛ ✶✳✸✿ ❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ❡①t❡r♥❛s ❞❡ Γ ❡ {O1 } = t1 ∩ t2 ✱ ♦s ♣♦♥t♦s A ❡ B ♦s ❝❡♥tr♦s ❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ P ❡ Q ♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞❡ t1 ❝♦♠ ♦s ❝ír❝✉❧♦s Γ ❞❡ s❡♠❡❧❤❛♥ç❛ AA t❡♠♦s q✉❡ △O1 AP ∼ △O1 BQ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ Ω✳ ❝ír❝✉❧♦s ❡ Ω✳ ▲♦❣♦✱ BQ O1 B = = k ⇒ BQ = k · AP. O1 A AP ❋✐❣✉r❛ ✶✳✹✿ ❍♦♠♦t❡t✐❛ ❣❡r❛❞❛ ♣❡❧❛s t❛♥❣❡♥t❡s ✐♥t❡r♥❛s ❞❡ ❋✐❣✉r❛ ✶✳✺✿ ❈ír❝✉❧♦s ❤♦♠♦tét✐❝♦s✳ Γ ❡ Ω✳ Γ ❡ Ω P❡❧♦ ❝❛s♦ ✶✽ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ Pr♦♣♦s✐çã♦ ✶✳✸✳ ❍♦♠♦t❡t✐❛s ❧❡✈❛♠ r❡t❛s ❡♠ r❡t❛s✳ r ✉♠❛ r❡t❛ ❡ Ho,k ❛ ❤♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k ✳ ❚♦♠❡ B ❡ C r✳ ′ ′ ❙❡❥❛♠ B = Ho,k (B) ❡ C = Ho,k (C)✳ P❡❧♦ ❝❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛✱ t❡♠♦s q✉❡ ♦s ′ ′ tr✐â♥❣✉❧♦s △OBC ❡ △OB C sã♦ s❡♠❡❧❤❛♥t❡s✳ ←→ ❙❡ P ❢♦r ✉♠ ♦✉tr♦ ♣♦♥t♦ q✉❛❧q✉❡r ❞❡ r = BC ✱ ❡♥tã♦✱ ♣❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦✱ ❝♦♥❝❧✉í♠♦s ←− → ′ ′ ′ ′ q✉❡ P = Ho,k (P ) ♣❡rt❡♥❝❡ à r❡t❛ B C ✳ P♦rt❛♥t♦ r = Ho,k (r) é ✉♠❛ r❡t❛✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ♣♦♥t♦s ❞❡ ❋✐❣✉r❛ ✶✳✻✿ ❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ r❡t❛ ❡♠ r❡t❛ ❞❡ r❛③ã♦ ❦❂✶✳✺ Pr♦♣♦s✐çã♦ ✶✳✹✳ ❍♦♠♦t❡t✐❛s ❧❡✈❛♠ ❝ír❝✉❧♦s ❡♠ ❝ír❝✉❧♦s✳ ❉❡♠♦♥str❛çã♦✳ ❉❡♥♦t❡ ♣♦r Ho,k A q✉❛❧q✉❡r ❞♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ k · OP O ❡ r❛③ã♦ k ✳ ❙❡❥❛♠ P ✉♠ ♣♦♥t♦ P ′ = Ho,k (P ) ❡ A′ = Ho,k (A)✳ ❊♥tã♦ OP ′ = ❛ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ ❡ r❛✐♦ r ❡ OA′ = k · OA✳ ′ ′ P❡❧♦ ❝❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛✱ ♦s tr✐â♥❣✉❧♦s △OAP ❡ △OA P sã♦ s❡♠❡❧❤❛♥t❡s✱ ❞❡ ♠♦❞♦ ′ ′ ′ ′ q✉❡ P A = k · r ✳ ❆ss✐♠✱ P ♣❡rt❡♥❝❡ ❛ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ A ❡ r❛✐♦ kr ✳ ❡ ❋✐❣✉r❛ ✶✳✼✿ ❍♦♠♦t❡t✐❛s ❧❡✈❛♥❞♦ ❝ír❝✉❧♦ ❡♠ ❝ír❝✉❧♦ ❞❡ r❛③ã♦ ❦❂✶✳✾ ✶✾ Pr♦♣♦s✐çã♦ ✶✳✺✳ ❍♦♠♦t❡t✐❛s ♣r❡s❡r✈❛♠ t❛♥❣ê♥❝✐❛ ❡♥tr❡ r❡t❛s ❡ ❝ír❝✉❧♦s✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ Ho,k ❛ ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ k ✱ r ✉♠❛ r❡t❛✱ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ A ❡ r❛✐♦ R✱ ❡ P ✉♠ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❡♥tr❡ Γ ❡ r✳ ❙❡ B ❡ C ♣❡rt❡♥❝❡ ❛ r ❡♥tã♦ r′ = Ho,k (r) é ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r B ′ = Ho,k (B) ❡ C ′ = Ho,k (C)✳ ❆❧é♠ ❞✐st♦✱ Γ′ = Hk,o (Γ) é ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ A′ ❡ r❛✐♦ kR✳ P♦r s❡♠❡❧❤❛♥ç❛ ❞❡ tr✐â♥❣✉❧♦s✱ ❝♦♥❝❧✉í♠♦s q✉❡ △OAP ∼ △OA′ P ′ ❡ △OP B ∼ △OP ′ B ′ ✳ π ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ ∠B ′ P ′ A′ = ∠BP A = ❞♦♥❞❡ s❡❣✉❡ q✉❡ P ′ é ✉♠ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ 2 ❡♥tr❡ Γ′ ❡ r′ ✳ ❋✐❣✉r❛ ✶✳✽✿ ❍♦♠♦t❡t✐❛ ❞❡ r❛③ã♦ ❦❂✶✳✾ ❉❡✜♥✐çã♦ ✶✳✷✳ ❙❡❥❛ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❙❡ ✉♠ ♣♦♥t♦ P ❡stá ❛ ✉♠❛ ❞✐stâ♥❝✐❛ d ❞❡ O✱ ❞❡✜♥✐♠♦s ❛ ♣♦tê♥❝✐❛ ❞♦ ♣♦♥t♦ P ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ Γ ❝♦♠♦ P ot(P ) = d2 − r2 . P❡❧❛ ❞❡✜♥✐çã♦ ❛♣r❡s❡♥t❛❞❛✱ s❡ P é ❡①t❡r✐♦r ❛ Γ✱ s✉❛ ♣♦tê♥❝✐❛ é ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦❀ s❡ P ♣❡rt❡♥❝❡ ❛ Γ✱ s✉❛ ♣♦tê♥❝✐❛ é ③❡r♦✱ s❡ P é ✐♥t❡r✐♦r ❛ Γ✱ s✉❛ ♣♦tê♥❝✐❛ é ♥❡❣❛t✐✈❛✳ ❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛♠ ❞❛❞♦s ✉♠ ❝ír❝✉❧♦ Γ✱ ❞❡ r❛✐♦ r ❡ ❝❡♥tr♦ O✱ ❡ ✉♠ ♣♦♥t♦ P ✳ ❙❡ ❛ r❡t❛ t q✉❡ ♣❛ss❛ ♣♦r P ❡ ✐♥t❡rs❡❝t❛ ♦ ❝ír❝✉❧♦ Γ ♥♦s ♣♦♥t♦s A ❡ B ✱ ❡♥tã♦ ♦ ♣r♦❞✉t♦ P A.P B é ✉♠❛ ❝♦♥st❛♥t❡ ✭✐st♦ é ✐♥❞❡♣❡♥❞❡ ❞❡ r✮✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ♦ ❝ír❝✉❧♦ Γ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✳ ❙❡❥❛ P ✉♠ ♣♦♥t♦ ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ Γ ❝♦♠ P O = d✳ ❙❡❥❛♠ t ✉♠❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s P ✱ A ❡ B ❡ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞♦ s❡❣♠❡♥t♦ AB ✳ ❙❡❥❛ ❛❣♦r❛ M A = M B = x✳ P❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ LLL✱ t❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s △AOM ❡ △BOM sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❈♦♥s❡q✉❡t❡♠❡♥t❡ ✱ ♦s â♥❣✉❧♦s ∠AM O ❡ ∠BM O sã♦ r❡t♦s✱ ♣♦✐s sã♦ ❝♦♥❣r✉❡♥t❡s ❡ s✉♣❧❡♠❡♥t❛r❡s✳ ❖✉ s❡❥❛✱ AM ⊥ OM ✳ P♦❞❡♠♦s ❞✐③❡r q✉❡✿ ✷✵ ❈❆P❮❚❯▲❖ ✶✳ ✶✳ ❙❡ P é ❡①t❡r✐♦r ❛♦ ❝ír❝✉❧♦ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ Γ✱ P A · P B = (P M − x)(P M + x) = P M 2 − x2 = P O2 − OM 2 − x2 = = P O2 − (OM 2 + x2 ) = d2 − r2 = P ot(P ). ❋✐❣✉r❛ ✶✳✾✿ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ ✷✳ ❙❡ P é ✐♥t❡r✐♦r ❛ P ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ Γ✱ −P A · P B = −(x − P M )(x + P M ) = P M 2 − x2 = P O2 − OM 2 − x2 = = P O2 − (OM 2 + x2 ) = d2 − r2 = P ot(P ). ❋✐❣✉r❛ ✶✳✶✵✿ P♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ P ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ ✷✶ ❖❜s❡r✈❛çã♦ ✶✳✸✳ ❙❡ P ♣❡rt❡♥❝❡ ❛ Γ✱ ❡♥tã♦ P s❡rá A ♦✉ B✱ ❞❡ ♠♦❞♦ q✉❡ pot(P ). ❖❜s❡r✈❛çã♦ ✶✳✹✳ ♥♦ ♣♦♥t♦ T ❡♥tã♦ ❙❡ P é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω P T 2 = P A · P B ✳ ❉❡ ❢❛t♦✱ ❈♦♥s✐❞❡r❡ ✉♠❛ r❡t❛ t t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω ♣♦✐s✱ ❝♦r❞❛ ❡ ✉♠❛ ❞❛s r❡t❛s é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ T ❡ ✉♠❛ r❡t❛ s q✉❡ ❝r✉③❛ ♦ △AT P ❡ △P T B s❡♠❡❧❤❛♥t❡s✱ ⌢ AT ✮✳ ♦✉ P T 2 = P A · P B. ❋✐❣✉r❛ ✶✳✶✶✿ ❙❡❣♠❡♥t♦ P❚ t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ ❉❡✜♥✐çã♦ ✶✳✸✳ Γ ✐♥s❝r✐t♦ ❛ ✉♠❛ ♠❡s♠❛ ▲♦❣♦✱ s❡✉s ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ♣r♦♣♦r❝✐♦♥❛✐s✱ ✐st♦ é✱ PA PT = PT PB ❛ C ♥✉♠ ♣♦♥t♦ Ω ♥♦s ♣♦♥t♦s A ❡ B ✳ ❚❡♠♦s ❡♥tã♦ ❞♦✐s tr✐â♥❣✉❧♦s ∠T P A = ∠T P B ❡ ∠AT P ∼ = ∠T BP ✭â♥❣✉❧♦s ❞❡ s❡❣♠❡♥t♦ ❝ír❝✉❧♦ PA·PB = 0 = ❙❡❥❛♠ ✉♠ ❝ír❝✉❧♦ s❡ ❡ s♦♠❡♥t❡ s❡ r Γ ❞❡ ❝❡♥tr♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ O ❡ ✉♠❛ r❡t❛ r✳ Ω✳ ❉✐③❡♠♦s q✉❡ Γ✳ ❋✐❣✉r❛ ✶✳✶✷✿ ❘❡t❛ ❡ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧✳ r é ♦rt♦❣♦♥❛❧ ✷✷ ❈❆P❮❚❯▲❖ ✶✳ ❋✐❣✉r❛ ✶✳✶✸✿ ❈ír❝✉❧♦ ❉❡✜♥✐çã♦ ✶✳✹✳ ♣♦♥t♦ I ❉♦✐s ❝ír❝✉❧♦s Γ ❡ Ω Γ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ♦rt♦❣♦♥❛❧ ❛♦ ❝ír❝✉❧♦ Ω sã♦ ❞✐t♦s ♦rt♦❣♦♥❛✐s s❡ ❡❧❡s s❡ ✐♥t❡rs❡❝t❛♠ ♥✉♠ ❢♦r♠❛♥❞♦ â♥❣✉❧♦s r❡t♦s✱ ✐st♦ é✱ s❡ s❡✉s r❛✐♦s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ♥♦ ♣♦♥t♦ I ❞❡ ✐♥t❡rs❡❝çã♦✳ ◆♦ ❝❛♣ít✉❧♦ ✸ ✉s❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛ s❡❣✉✐r✳ Pr♦♣♦s✐çã♦ ✶✳✻✳ ❯♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ s❡✱ ❡ só s❡✱ ❞✉❛s ❞✐❛❣♦♥❛✐s s❡ ✐♥t❡rs❡❝t❛♠ ♥♦s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s ♠é❞✐♦s✳ M ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝✲ ∼ çã♦ ❞❡ s✉❛s ❞✐❛❣♦♥❛✐s✳ ❉❡ AB k CD ✱ s❡❣✉❡ q✉❡ ∠BAM = ∠DCM ❡ ∠ABM ∼ = ∠CDM ✳ ❈♦♠♦ ❥á s❛❜❡♠♦s q✉❡ AB = CD ✱ s❡❣✉❡ q✉❡ ♦s tr✐â♥❣✉❧♦s △ABM ❡ △CDM sã♦ ❝♦♥❣r✉✲ ❡♥t❡s ♣❡❧♦ ❝❛s♦ ALA✳ ▲♦❣♦✱ AM = CM ❡ BM = DM ✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ ABCD ✉♠ q✉❛❞r✐❧át❡r♦ t❛❧ q✉❡ s✉❛s ❞✐❛❣♦♥❛✐s AC ❡ BD s❡ ✐♥t❡rs❡❝✲ t❛♠ ❡♠ M ✱ ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ❛♠❜❛s✳ ❊♥tã♦ M A = M C ✱ BM = DM ❡ ∠AM B ∼ = ∠CM D ✭â♥❣✉❧♦s ♦♣st♦s ♣❡❧♦ ✈ért✐❝❡✮✱ ❞❡ ♠♦❞♦ q✉❡ ♦s tr✐â♥❣✉❧♦s △ABM ❡ △CDM sã♦ ❝♦♥❣r✉✲ ❡♥t❡s✱ ♣♦r LAL✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ △BCM ❡ △DAM t❛♠❜é♠ sã♦ ❝♦♥❣r✉❡♥t❡s ♣♦r LAL✳ ❚❛✐s ❝♦♥❣r✉ê♥❝✐❛s ♥♦s ❞ã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ AB = CD ❡ BC = AD ✱ ♦ q✉❡ ❥á s❛❜❡♠♦s s❡r ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❢❛t♦ ❞❡ ABCD s❡r ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ s❡❥❛ ABCD ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡ ❋✐❣✉r❛ ✶✳✶✹✿ ❆s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ✷✸ Pr♦♣♦s✐çã♦ ✶✳✼✳ ❙❡ ✉♠ tr✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ♥✉♠ s❡♠✐❝ír❝✉❧♦ Γ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r t❡♠ ✉♠ ❧❛❞♦ ❝✉❥❛ ♠❡❞✐❞❛ é ✐❣✉❛❧ ❛♦ s❡✉ ❞✐â♠❡tr♦✱ ❡♥tã♦ ❡❧❡ é ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❡ ❡ss❡ ❞✐â♠❡tr♦ é ❤✐♣♦t❡♥✉s❛ ❞♦ tr✐â♥❣✉❧♦✳ ❙❡❥❛♠ D✱ A ❡ B ♣♦♥t♦s ❞❡ Γ t❛✐s q✉❡ A✱ O ❡ B s❡❥❛♠ ❝♦❧✐♥❡❛r❡s✳ t❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ △BOD é ✐sós❝❡❧❡s✱ ❞❡ ♠♦❞♦ q✉❡ ∠ODB ∼ = ∠OBD✱ ❞❡ ♠❡❞✐❞❛ α✳ ❆♥á❧♦❣❛♠❡♥t❡✱ ∠OAD ∼ = ∠ODA✱ ❞❡ ♠❡❞✐❞❛ β ✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❉❡♠♦♥str❛çã♦✳ 2α + 2β = π. π 2 ▲♦❣♦✱ α + β = ✳ ❆ss✐♠✱ ∠ADB ♠❡❞❡ π ❡ ♦ tr✐â♥❣✉❧♦ △ADB é r❡tâ♥❣✉❧♦ ❡♠ D✳ 2 ❋✐❣✉r❛ ✶✳✶✺✿ ❚r✐â♥❣✉❧♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦ ❆ ❞❡✜♥✐çã♦ ❞❡ ❡❧✐♣s❡ s❡rá ♥❡❝❡ssár✐❛ ♣❛r❛ ❢❛❧❛r♠♦s ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s q✉❡ s❡rá ❛❜♦r❞❛❞❛ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✳ ❉❡✜♥✐çã♦ ✶✳✺✳ ❋✐①❛❞♦ ❞♦✐s ♣♦♥t♦s F1 ❡ F2 ❞♦ ♣❧❛♥♦ ✉♠❛ ❡❧✐♣s❡ ξ ❞❡ ❢♦❝♦s é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ ❝✉❥❛ s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s ❛ F1 2a > 0✱ d(F1 , F2 ) = 2c✱ ❖✉ s❡❥❛✱ ❝♦♥st❛♥t❡ ♠❛✐♦r ❞♦ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❢♦❝♦s 2c ≥ 0✳ ξ = {P |d(P, F1 ) + d(P, F2 ) = 2a} ❋✐❣✉r❛ ✶✳✶✻✿ ❊❧✐♣s❡ ❞❡ ❢♦❝♦s F1 ❡ F2 ✳ ❡ F2 F1 ❡ F2 é ✐❣✉❛❧ ✉♠❛ 0≤c 0 ❡ ♠♦str❛♠♦s ❝♦♠♦ ✐♥✈❡rt❡r r❡t❛s ❡ ❝ír❝✉❧♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❞❛❞♦ ❝ír❝✉❧♦✳ ❆ ✐♥✈❡rsã♦ é ✉♠❛ té❝♥✐❝❛ ✉s❛❞❛ ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦ ♥♦ ❡st✉❞♦ ❞♦s ❛r❜❡❧♦s✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ ❝♦♥str✉çã♦ ❞❛s ❝❛❞❡✐❛s ❞❡ P❛♣♣✉s ❡ ❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ✷✳✶ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r > 0 ✜①❛❞♦s✱ ♥♦ ♣❧❛♥♦✳ ❆ ✐♥✈❡rsã♦ I ❡♠ r❡❧❛çã♦ ❛ Γ é ❞❡✜♥✐❞❛ ❝♦♠♦ ❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣♦♥t♦ P ✱ ❞✐st✐♥t♦ ❞❡ O✱ −→ ♥♦ ♣❧❛♥♦ ✉♠ ú♥✐❝♦ ♣♦♥t♦ P ′ ♣❡rt❡♥❝❡♥t❡ à s❡♠✐rr❡t❛ OP t❛❧ q✉❡ OP · OP ′ = r2 ✳ ❋✐❣✉r❛ ✷✳✶✿ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦✳ ❆ s❡❣✉✐r✱ ❞❡t❡r♠✐♥❛♠♦s ✉♠❛ ❡①♣r❡ssã♦ ❛❧❣é❜r✐❝❛ ♣❛r❛ ❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ Γ✳ ❈♦♠❡ç❛♠♦s s✉♣♦♥❞♦ q✉❡ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ✷✼ ✷✽ ❈❆P❮❚❯▲❖ ✷✳ ●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆ ✶◦ ❝❛s♦✿ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ✐st♦ é✱ O = (0, 0)✳ ❋✐❣✉r❛ ✷✳✷✿ O é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❙❡❥❛♠ P = (x, y) 6= (0, 0) ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ♣❧❛♥♦ ❡ P ′ = (x′ , y ′ ) ❛ ✐♠❛❣❡♠ ❞❡ P −→ ♣❡❧❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ Γ✱ ✐st♦ é✱ P ′ = I(P )✳ ❈♦♠♦ P ′ ∈ OP ✱ ❡♥tã♦ (x′ , y ′ ) = λ(x, y), ❝♦♠ λ ∈ R+ ∗. p p P♦r ❞❡✜♥✐çã♦✱ OP · OP ′ = r2 ✱ ❞♦♥❞❡ t❡♠♦s q✉❡ x2 + y 2 · (x′ )2 + (y ′ )2 = r2 ✳ r2 r2 = ✳ P♦rt❛♥t♦✱ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ λ2 · (x2 + y 2 )2 = r4 ✳ ▲♦❣♦✱ λ = 2 x + y2 (OP )2 I(x, y) = P ′ = −→ r2 · OP . 2 (OP ) ❈❛s♦ ❣❡r❛❧✿ O = (x0 , y0 ) é ❞✐st✐♥t♦ ❞❡ (0, 0)✿ ◆♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s ♦r✐❣✐♥❛❧✱ s✉♣♦♥❤❛ q✉❡ P t❡♥❤❛ ❝♦♦r❞❡♥❛❞❛s P = (x1 , y1 )✳ ◆♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s q✉❡ t❡♠ O = (x0 , y0 ) ❝♦♠♦ ♦r✐❣❡♠✱ ❛s ❝♦♦r❞❡✲ ♥❛❞❛s ❞❡ P sã♦ P = (x1 − x0 , y1 − y0 )✳ ❊♠ r❡❧❛çã♦ ❛ ❡st❡ s✐st❡♠❛✱ ❛s ❝♦♦r❞❡♥❛❞❛s ❞♦ −→ r2 · OP ✳ P❛r❛ ♦❜t❡r ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ P ′ ♥♦ 2 (OP ) s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♦r✐❣✐♥❛❧✱ ♣r❡❝✐s❛♠♦s tr❛♥s❧❛❞❛r ♦s ❡✐①♦s ❞❡ ♠♦❞♦ q✉❡ (0, 0) s❡❥❛ ♥♦✈❛♠❡♥t❡ ❛ ♦r✐❣❡♠✳ ■st♦ é ❢❡✐t♦ s♦♠❛♥❞♦✲s❡ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡ O às ❝♦♦r❞❡♥❛❞❛s ♥♦ ♣♦♥t♦ P ′ = I(P ) sã♦ ❞❛❞❛s ♣♦r P ′ = ♦✉tr♦ s✐st❡♠❛✳ ❖✉ s❡❥❛✱ (x′ , y ′ ) = −→ r2 · OP + O. (OP )2 ✷✳✶✳ ✷✾ ■◆❱❊❘❙➹❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼ ❈❮❘❈❯▲❖ ❋✐❣✉r❛ ✷✳✸✿ O ♥ã♦ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✳ ❊♠ ❝♦♦r❞❡♥❛❞❛s t❡♠♦s✱ I(x, y) = (x′ , y ′ ) = (x0 , y0 ) + r2 · (x1 − x0 , y1 − y0 ). (x1 − x0 )2 + (y1 − y0 )2 ❉❡st❛ ❢♦r♠❛✱ ❛ ❡①♣r❡ssã♦ ❣❡r❛❧ ♣❛r❛ ❛ ✐♥✈❡rsã♦ I : R2 \ {O} −→ R2 \ {O} ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ Γ ❞❡ ❝❡♥tr♦ O é r2 −→ · OP . I(P ) = O + OP 2 ✭✷✳✶✮ ❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ❛❧❣✉♠❛s ❝♦♥s❡q✉ê♥❝✐❛s ✐♠❡❞✐❛t❛s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥✈❡rsã♦✿ −−→ ✶✳ OP ′ = r2 −→ · OP ✱ ❞❡ ♠♦❞♦ q✉❡ OP ′ · OP = r2 ✳ OP 2 ✷✳ ❙❡ P ❢♦r ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r ❞❡ Γ✱ ✐st♦ é✱ OP > r ❡♥tã♦ r2 = OP · OP ′ > r · OP ′ ❀ ❧♦❣♦✱ OP ′ < r✱ ✐st♦ é✱ P ′ ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ Γ✳ ❋✐❣✉r❛ ✷✳✹✿ ❖ ♣♦♥t♦ P é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Γ✳ ✸✵ ❈❆P❮❚❯▲❖ ✷✳ ❋✐❣✉r❛ ✷✳✺✿ ❖ ♣♦♥t♦ ✸✳ ❙❡ P ❧♦❣♦✱ ✹✳ ❙❡ P ❢♦r ✉♠ ♣♦♥t♦ ❞♦ ✐♥t❡r✐♦r ❞❡ OP > r✱ ′ ✐st♦ é✱ P ′ ❢♦r ✉♠ ♣♦♥t♦ ❞❡ ❞❡ ♦✉ s❡❥❛✱ P ′ ∈ Γ✳ Γ✱ I Γ✳ é ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ OP < r ❞❡ Γ✳ ✐st♦ é✱ ❡stá ♥♦ ❡①t❡r✐♦r ❡♥tã♦ r2 = OP · OP < r · OP ❀ Γ✱ ✐st♦ é✱ OP = r ❡♥tã♦ r2 = r ·OP ′ ✱ ❞❡ ♠♦❞♦ q✉❡✱OP ′ = r✱ ❋✐❣✉r❛ ✷✳✻✿ ❖ ♣♦♥t♦ ✺✳ P ●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆ P ♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦ Γ✳ é ✉♠❛ ✐♥✈♦❧✉çã♦✱ ✐st♦ é✱ −−→′ r2 r2 · ( OP ) = O + I(I(P )) = I(P ) = O +  2 2 · (OP ′ )2 r OP ′   −→ r2 −→ = O + OP = P. 2 · OP OP ✷✳✷✳ ✸✶ ■◆❱❊❘❙➹❖ ❉❆ ❘❊❚❆ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼ ❈❮❘❈❯▲❖ ✷✳✷ ■♥✈❡rsã♦ ❞❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ Γ Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡❥❛ ✉♠❛ r❡t❛ s O q✉❡ ♣❛ss❛ ♣♦r ✉♠ ♣♦♥t♦ ❞❛ r❡t❛ ♣❡rt❡♥❝❡♠ à s✱ é ❛ ♣ró♣r✐❛ r❡t❛ O ❡ r❛✐♦ r✳ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ❡♥tã♦ P ′ s✱ ❡♥tã♦ ♦ ✐♥✈❡rs♦ ♣❡rt❡♥❝❡ ❛ Γ ❞❡ s✳ ❱❛♠♦s ♠♦str❛r q✉❡ ❛s ✐♥✈❡rs❛s ❞♦s ♣♦♥t♦s ❞❡ ❉❡♠♦♥str❛çã♦✳ P 6= O ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ P ′ s s✳ ❈♦♠♦ P t❛♠❜é♠ ❡stã♦ ❡♠ ♣❡rt❡♥❝❡ à s❡♠✐rr❡t❛ −→ OP ✳ ❙❡❥❛ ❡ O s✳ ❆❣♦r❛✱ r❡st❛ ♠♦str❛r q✉❡ ❝❛❞❛ ♣♦♥t♦ B ❞❡ s é ✐♥✈❡rs♦ ❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞❡ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❢❛t♦ ❞❡ ❛ ✐♥✈❡rsã♦ s❡r ✉♠❛ ✐♥✈♦❧✉çã♦✳ ❖✉ s❡❥❛ s✳ ■st♦ é Q = (I(Q))✳ ❋✐❣✉r❛ ✷✳✼✿ ■♥✈❡rsã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳ Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡❥❛ ✉♠❛ r❡t❛ s Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ q✉❡ ♥ã♦ ♣❛ss❛ ♣♦r ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ s O O ❡ r❛✐♦ r✳ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ é ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣♦r ❛ r❡t❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ Γ ❞❡ O✳ O ❞♦ ❝ír❝✉❧♦ Γ ❡ t ✉♠ r❡t❛ s q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ O ❞❡ Γ ❡ ✐♥t❡rs❡❝t❛ ❛ r❡t❛ s ♥♦ ♣♦♥t♦ A✳ ❙❡❥❛ A′ ♦ ✐♥✈❡rs♦ ❞♦ ♣♦♥t♦ A ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦✳ ❋✐①❡ ✉♠ ♣♦♥t♦ P ❞❡ s✱ P 6= A✱ ❡ s❡❥❛ P ′ ♦ s❡✉ ✐♥✈❡rs♦ ❡♠ r❡❧❛çã♦ ❛ Γ✳ ❚❡♠♦s q✉❡ OA · OA′ = OP · OP ′ = r2 ✳ P❡❧♦ ❝❛s♦ LAL ❞❡ s❡♠❡❧❤❛♥ç❛ t❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s △AOP ❡ △A′ OP ′ sã♦ s❡♠❡❧❤❛♥t❡s✳ ❆ss✐♠✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✭✶✳✼✮✱ ❝♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ ♦ â♥❣✉❧♦ ∠OP ′ A′ é r❡t♦✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❝♦♥❝❧✉✐r q✉❡ ♦ s❡❣♠❡♥t♦ OA′ é ♦ ❞✐â♠❡tr♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ O✳ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ ❛❣♦r❛ ♦ ♣♦♥t♦ ✸✷ ❈❆P❮❚❯▲❖ ✷✳ ●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆ ❋✐❣✉r❛ ✷✳✽✿ ❆ r❡t❛ s é ❡①t❡r♥❛ ❛♦ ❝ír❝✉❧♦ Γ✳ ❋✐❣✉r❛ ✷✳✾✿ ❆ r❡t❛ s é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ❋✐❣✉r❛ ✷✳✶✵✿ ❆ r❡t❛ s é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Γ✳ ✷✳✸✳ ■◆❱❊❘❙➹❖ ❉❊ ❯▼ ❈❮❘❈❯▲❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼ ❈❮❘❈❯▲❖ ❉❆❉❖ ✷✳✸ ✸✸ ■♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ❞❛❞♦ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O′ ✱ ❛ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ Γ ❞❡ ✉♠ ❝ír❝✉❧♦ r q✉❡ ♥ã♦ ♣❛ss❛ ♣♦r O′ é ✉♠ ❝ír❝✉❧♦✳ Pr♦♣♦s✐çã♦ ✷✳✸✳ ❙❡❥❛ Ω ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ ❉❡♠♦♥str❛çã♦✳ ✶ ◦ ❝❛s♦✿ O ❡stá ♥♦ ❡①t❡r✐♦r ❞❡ Γ✳ ❚r❛❝❡ OA t❛♥❣❡♥t❡ ❛ Γ ♥♦ ♣♦♥t♦ A✳ P❛r❛ ❝❛❞❛ ♣♦♥t♦ B ♣❡rt❡♥❝❡♥t❡ ❛ Γ✱ ❞❡ ♠♦❞♦ q✉❡ OB ♥ã♦ s❡❥❛ t❛♥❣❡♥t❡ ❛ Γ✱ ❝♦♥s✐❞❡r❡ B ′ ♦ ♦✉tr♦ ♣♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❡ Γ ❝♦♠ ❛ s❡♠✐rr❡t❛ −−→ OB ✳ P❡❧❛ ♦❜s❡r✈❛çã♦ ✭✶✳✹✮✱ t❡♠✲s❡ q✉❡ OB · OB ′ = OA2 . ❙❡❥❛ ❛❣♦r❛ B ′′ ♦ ✐♥✈❡rs♦ ❞❡ B′ ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ Ω✳ ✭✷✳✷✮ ❊♥tã♦✱ OB ′ · OB ′′ = r2 ✭✷✳✸✮ ❉❡ ✭✷✳✷✮ ❡ ✭✷✳✸✮✱ t❡♠♦s q✉❡  r 2 · OB OB = OA ▲♦❣♦ ♦ ✐♥✈❡rs♦ ❞♦ ❝ír❝✉❧♦ Γ✱ ❡♠ r❡❧❛çã♦ ❛ Ω✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✭✶✳✹✮✱ é s✉❛ ✐♠❛❣❡♠ ♣❡❧❛  r 2 ❤♦♠♦t❡t✐❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ ✳ OA ❆ss✐♠✱ ❛ ✐♥✈❡rsã♦ ❧❡✈❛ ♦ ❝ír❝✉❧♦ Γ✱ ❝♦♠ O ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ Γ✱ ♥✉♠ ❝ír❝✉❧♦ Γ′ ✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ t♦❞♦ ♣♦♥t♦ ❞♦ ❝ír❝✉❧♦ Γ′ é ♦ ✐♥✈❡rs♦✱ r❡❧❛t✐✈♦ ❛♦ ❝ír❝✉❧♦ Ω ❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞❡ Γ✳ ′′ ❋✐❣✉r❛ ✷✳✶✶✿ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ ✷ ❙❡❥❛ ◦ ❝❛s♦✿ B O ❡stá ♥♦ ✐♥t❡r✐♦r ❞❡ Γ é ❡①t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω✳ Γ✳ ✉♠ ♣♦♥t♦ q✉❡ ♣❡rt❡♥❝❡ ❛♦ ❝ír❝✉❧♦ Γ ❡ B′ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ ←→ OB ❝♦♠ ✸✹ ❈❆P❮❚❯▲❖ ✷✳ ●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆ ❋✐❣✉r❛ ✷✳✶✷✿ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ❋✐❣✉r❛ ✷✳✶✸✿ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ Ω✳ ♦ ❝ír❝✉❧♦ Γ ❡ ❝♦♥s✐❞❡r❡ k = OB · OB ′ ✳ ❆❣♦r❛ ♣❛r❛ ❝❛❞❛ A 6= B ♣❡rt❡♥❝❡♥t❡ ❛ Γ✱ s❡❥❛ A′ ♦ ♦✉tr♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ Γ ❝♦♠ ←→ ❛ r❡t❛ OA✳ P❡❧❛ ♣♦tê♥❝✐❛ ❞❡ ✉♠ ♣♦♥t♦ t❡♠♦s q✉❡✱ OB · OB ′ = OA · OA′ = k ✳ ❙❡ A′′ é ♦ ✐♥✈❡rs♦ ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ Ω ❞♦ ♣♦♥t♦ A′ ✱ q✉❡ ♣❡rt❡❝❡ ❛ Γ✱ ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ Ω✱ ❡♥tã♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥✈❡rsã♦ t❡♠♦s q✉❡ OA′ · OA′′ = r2 ✱ ♦♥❞❡ r é ♦ r❛✐♦ ❞❡ Ω✳ r2 ▲♦❣♦ OA′′ = · OA✳ k ❆ss✐♠✱ ❛ ✐♥✈❡rsã♦✱ ❡♠ r❡❧❛çã♦ ❛ Ω✱ ❧❡✈❛ Γ ♥❛ s✉❛ ✐♠❛❣❡♠ ❤♦♠♦tét✐❝❛ ❞❡ ❝❡♥tr♦ O ❡ r❛③ã♦ r2 ✱ ♦♥❞❡ k = OB · OB ′ ❡✱ ♣♦rt❛♥t♦✱ ♥✉♠ ❝ír❝✉❧♦ Γ′ ✳ k ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ t♦❞♦ ♣♦♥t♦ ❞♦ ❝í❝✉❧♦ Γ′ é ♦ ✐♥✈❡rs♦✱ r❡❧❛t✐✈♦ ❛♦ ❝ír❝✉❧♦ Ω✱ ❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞❡ Γ✳ ✷✳✸✳ ■◆❱❊❘❙➹❖ ❉❊ ❯▼ ❈❮❘❈❯▲❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼ ❈❮❘❈❯▲❖ ❉❆❉❖ ❋✐❣✉r❛ ✷✳✶✹✿ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ ❋✐❣✉r❛ ✷✳✶✺✿ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ Γ Γ ❋✐❣✉r❛ ✷✳✶✻✿ ❖ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ é s❡❝❛♥t❡ ❛♦ ❝ír❝✉❧♦ é t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Γ ✐♥t❡r♥♦ ❛♦ ❝ír❝✉❧♦ Ω✳ Ω✳ Ω✳ ✸✺ ✸✻ ❈❆P❮❚❯▲❖ ✷✳ Pr♦♣♦s✐çã♦ ✷✳✹✳ ❙❡❥❛ ✉♠ ❝ír❝✉❧♦ Ω Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ q✉❡ ♣❛ss❛ ♣♦r O O ❡ r❛✐♦ r✳ ●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆ ❆ ✐♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ é ✉♠❛ r❡t❛ q✉❡ ♥ã♦ ♣❛ss❛ ♣♦r Γ ❞❡ O✳ ❙❡❥❛ Γ ✉♠ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r ❡ Ω ✉♠ ❝ír❝✉❧♦ ❞❡ ←−→ ❝❡♥tr♦ O1 q✉❡ ♣❛ss❛ ♣♦r O✳ ❙❡❥❛ A 6= O ♦ ♣♦♥t♦ ♦♥❞❡ ❛ r❡t❛ s = OO1 ✐♥t❡rs❡❝t❛ Ω✳ ❙❡❥❛ A′ ♦ ✐♥✈❡rs♦ ❞❡ A ❡♠ r❡❧❛çã♦ ❛ Γ✱ ❡♥tã♦✱ A′ ♣❡rt❡♥❝❡ ❛ s✳ ❙❡❥❛ r ❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r A′ ❡ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ s✳ Pr♦✈❛r❡♠♦s q✉❡ r = Ω′ ✱ ♦✉ s❡❥❛✱ ❛ ✐♥✈❡rsã♦ ❞♦ ❝ír❝✉❧♦ Ω ❡♠ r❡❧❛çã♦ ❛ Γ é ❛ r❡t❛ r✳ ←→ ❙❡❥❛♠ ❛❣♦r❛ B 6= A ❡ B 6= O ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞♦ ❝ír❝✉❧♦ Ω ❡ t = OB ✳ ❆❣♦r❛✱ s❡❥❛ P ♦ ♣♦♥t♦ ♦♥❞❡ r ✐♥t❡rs❡❝t❛ t✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ P = B ′ ✳ ❚❡♠♦s q✉❡ ♦s tr✐â♥❣✉❧♦s △ABO ❡ △OA′ P sã♦ s❡♠❡❧❤❛♥t❡s✱ ♣♦✐s✱ ♦s â♥❣✉❧♦s ∠A′ OP = ∠AOB sã♦ ❝♦♠✉♥s ❡ ♦ â♥❣✉❧♦ ∠ABO = π/2✱ ♣♦✐s✱ OA é ♦ ❞✐â♠❡tr♦ ❞♦ ❝ír❝✉❧♦ ❡ ✉♠ ❞♦s ❉❡♠♦♥str❛çã♦✳ OB OA ❧❛❞♦s ❞♦ tr✐â♥❣✉❧♦ △ABO ❡ ∠OA′ P = π/2✱ ♣♦r ❝♦♥str✉çã♦✳ ❊♥tã♦✱ = ✳ OA′ OP ▲♦❣♦✱ OB · OP = OA · OA′ ✱ ♠❛s ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✐♥✈❡rsã♦ s❛❜❡♠♦s q✉❡ OA · OA′ = r2 ✳ ❈♦♥❝❧✉í♠♦s q✉❡ OB · OP = r2 ✱ s❡♥❞♦ P ♦ ✐♥✈❡rs♦ ❞❡ B ✱ ♦✉ s❡❥❛ P = B ′ ✳ P♦rt❛♥t♦✱ ♦s ✐♥✈❡rs♦s ❞♦s ♣♦♥t♦s B 6= O ❞♦ ❝ír❝✉❧♦ Ω ♣❡rt❡♥❝❡♠ ❛ r❡t❛ r✱ ♦✉ s❡❥❛✱ ❛ r❡t❛ r é ❛ ✐♥✈❡rs❛ ❞❡ Ω ❡♠ r❡❧❛çã♦ ❛ Γ✳ ❋✐❣✉r❛ ✷✳✶✼✿ ■♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳ Pr♦♣♦s✐çã♦ ✷✳✺✳ ❆ ✐♥✈❡rsã♦ ❞❡ ✉♠ ❝ír❝✉❧♦ é ♦ ♣ró♣r✐♦ ❝ír❝✉❧♦ Ω ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ Γ ♦rt♦❣♦♥❛❧ ❛ Ω✱ Ω✳ ❈♦♥s✐❞❡r❡ ✉♠ ❝ír❝✉❧♦ Γ✱ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r✱ ❡ ✉♠ ❝ír❝✉❧♦ Ω q✉❡ ✐♥t❡rs❡❝t❛ ←→ Γ ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ♥♦s ♣♦♥t♦s A ❡ B ✳ ▲♦❣♦✱ ❛ r❡t❛ s = OA s❡rá t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ Ω ❉❡♠♦♥str❛çã♦✳ ✷✳✸✳ ■◆❱❊❘❙➹❖ ❉❊ ❯▼ ❈❮❘❈❯▲❖ ❊▼ ❘❊▲❆➬➹❖ ❆ ❯▼ ❈❮❘❈❯▲❖ ❉❆❉❖ E ♣❡rt❡♥❝❡♥t❡ ❛ C ❡ ❛ s❡♠✐rr❡t❛ ♣❛ss❛♥❞♦ ♣♦r O ❡ E ′ q✉❡ ✐♥t❡rs❡❝t❛ ♦ ❝ír❝✉❧♦ Ω ♥♦ ♣♦♥t♦ E ✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣♦tê♥❝✐❛ ❞♦ ♣♦♥t♦ O ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ Ω✱ t❡♠♦s✱ (OA)2 = OE · OE ′ = r2 , ❡♠ A✳ ✸✼ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ✉♠ ♣♦♥t♦ ♦ q✉❡ ❝❛r❛❝t❡r✐③❛ q✉❡ E′ r❡❛❧♠❡♥t❡ é ♦ ✐♥✈❡rs♦ ❞❡ ❝ír❝✉❧♦ ♣❛ss❛♥❞♦ ♣❡❧♦s ♣♦♥t♦s A✱ B ❡ E✳ ′ E ❡♠ r❡❧❛çã♦ ❛ ▲♦❣♦✱ ♦ ✐♥✈❡rs♦ ❞❡ Ω Γ✳ ▼❛s✱ ❤á ❛♣❡♥❛s ✉♠ ❡♠ r❡❧❛çã♦ ❛ ❋✐❣✉r❛ ✷✳✶✽✿ ■♥✈❡rsã♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❝ír❝✉❧♦ ♦rt♦❣♦♥❛❧ Γ é Ω✳ ✸✽ ❈❆P❮❚❯▲❖ ✷✳ ●❊❖▼❊❚❘■❆ ■◆❱❊❘❙■❱❆ ❈❛♣ít✉❧♦ ✸ ❆r❜❡❧♦s ◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ♦s ❛r❜❡❧♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐✲ ♠❡❞❡s✱ ❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ❡ ♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ❉❡✜♥✐çã♦ ✸✳✶✳ C1 ❡ C3 ✱ ❉❛❞♦s ✸ ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s C1 ✱ C2 ❡ C3 s♦❜r❡ ✉♠❛ r❡t❛ ♦ ❛r❜❡❧♦ é ❛ r❡✉♥✐ã♦ ❞❡ três s❡♠✐❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s ✉♠ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ q✉❡ ❝♦♥té♠ C1 ✱ C2 ❡ C1 C2 ✱ r✱ ❝♦♠ C2 ❡♥tr❡ C2 C3 ❡ C1 C3 ❞❡ C3 ✳ ❋✐❣✉r❛ ✸✳✶✿ ❆r❜❡❧♦s ❆ ✜❣✉r❛ ❛ s❡❣✉✐r s❡rá ✉t✐❧✐③❛❞❛ ♣❛r❛ ✐❧✉str❛r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❛r❜❡❧♦s✳ ❆s ♠❡s♠❛s ♣♦❞❡♠ s❡r ♠♦str❛❞❛s ✉s❛♥❞♦✲s❡ ❛ ❣❡♦♠❡tr✐❛ ❡st✉❞❛❞❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳ Pr♦❝✉✲ r❛r❡♠♦s ♠♦str❛r ❛❧❣✉♠❛s ❞❡❧❛s ❞❡ ♠❛♥❡✐r❛ ❜❡♠ ❞❡t❛❧❤❛❞❛ ❞❡ ♠♦❞♦ ❛ ❛❥✉❞❛r ♦ ❛❧✉♥♦ ❛ ✸✾ ✹✵ ❈❆P❮❚❯▲❖ ✸✳ ❆❘❇❊▲❖❙ t❡r ✉♠ ❜♦♠ ❡♥t❡♥❞✐♠❡♥t♦ s♦❜r❡ ♦ ❛ss✉♥t♦✳ C2 ✱ ❝♦♥str✉✐♠♦s ✉♠❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ C1 C3 q✉❡ ✐♥t❡rs❡❝t❛ ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C1 C3 ♥♦ ♣♦♥t♦ D ✳ ❈♦♥str✉í♠♦s ♦ s❡❣♠❡♥t♦ DC1 ✳ ❙❡❥❛ F ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C1 C2 ❝♦♠ ♦ s❡❣♠❡♥t♦ DC1 ✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ ❝♦♥str✉í♠♦s DC3 q✉❡ ❝♦rt❛rá ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C2 C3 ♥♦ ♣♦♥t♦ H ✳ ◆♦ ♣♦♥t♦ ❋✐❣✉r❛ ✸✳✷✿ ❆r❜❡❧♦s Pr♦♣♦s✐çã♦ ✸✳✶✳ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡♠✐❝ír❝✉❧♦ s✉♣❡r✐♦r ❞❡ ✉♠ ❛r❜❡❧♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s s❡♠✐❝ír❝✉❧♦s ✐♥❢❡r✐♦r❡s ❞♦ ❛r❜❡❧♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ α ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ ♠❡♥t♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C2 C3 ✳ ❡♥tã♦ C1 C2 ❡ γ ♦ ❝♦♠♣r✐✲ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡ C1 C3 = 1 ✳ C2 C3 = 1 − r ✳ ▲♦❣♦✱ t❡♠♦s q✉❡✿ 1 2 = π, α = 2 2 r 2·π· 2 = πr , β = 2 2 1−r 2·π· 2 = π − πr . γ = 2 2 2·π· C1 C3 ✱ β ❙❡ C1 C2 = r ✱ ❝♦♠ 0 < r < 1✱ ✹✶ ❆ss✐♠✱ t❡♠♦s q✉❡✿   π − πr πr + β+γ = 2 2 πr π πr π + − = = α. 2 2 2 2 Pr♦♣♦s✐çã♦ ✸✳✷✳ ❖ q✉❛❞r✐❧át❡r♦ C2 F DH ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ ♦ tr✐â♥❣✉❧♦ C1 C3 é ✉♠ r❡tâ♥❣✉❧♦✳ △C1 DC3 ❡stá ✐♥s❝r✐t♦ ❡♠ ✉♠ s❡♠✐❝ír❝✉❧♦ ❡ s❡✉ ❧❛❞♦ △C1 DC3 é r❡tâ♥❣✉❧♦ ❡♠ D✳ ∠C2 C3 D = a ❡ ♦ â♥❣✉❧♦ ∠C2 C1 D = b✱ é ♦ ❞✐â♠❡tr♦ ❞❡ss❡ s❡♠✐❝ír❝✉❧♦ t❡♠♦s q✉❡ ♦ tr✐â♥❣✉❧♦ ∠C1 DC3 = π/2✳ ❙❡❥❛ ♦ â♥❣✉❧♦ a + b = π/2✳ ❖ tr✐â♥❣✉❧♦ △DC2 C3 é✱ ♣♦r ❝♦♥str✉çã♦✱ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ♣♦✐s C2 D é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ C1 C3 ✳ ❈♦♠♦ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ ❛ π ✱ ❡ ♦ â♥❣✉❧♦ ∠C2 C3 D = a✱ ❡♥tã♦ ♦ â♥❣✉❧♦ ∠C2 DC3 = b✳ ❈♦♠♦ ♦s tr✐â♥❣✉❧♦s △C1 F C2 ❡ △C2 HC3 ❡stã♦ ✐♥s❝r✐t♦s ❡♠ s❡♠✐❝ír❝✉❧♦s ❡ ✉♠ ❞❡ s❡✉s ❧❛❞♦s é ❞✐â♠❡tr♦ ❞❡st❡s s❡♠✐❝ír❝✉❧♦s✱ ❡♥tã♦ ❡❧❡s sã♦ tr✐â♥❣✉❧♦s r❡tâ♥❣✉❧♦s✳ ▲♦❣♦✱ ∠C1 F C2 = ∠C2 HC3 = π/2✳ ❯♠❛ ✈❡③ q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ ❛ π ✱ s❡❣✉❡ q✉❡ ♦ â♥❣✉❧♦ ∠F C2 C1 = a ❡ ♦ â♥❣✉❧♦ ∠HC2 C3 = b✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ∠C1 C2 F = a ❡ ∠C2 C3 H = b✳ ❈♦♠♦ ∠DC2 C1 = ∠DC2 C3 = π/2✱ ❡♥tã♦ ∠DC2 F = π/2 − a = b ❡ ∠DC2 H = π/2 − b = a✳ P❡❧♦ ❝❛s♦ ALA ♦s tr✐â♥❣✉❧♦s △C2 F D ❡ △C2 HD sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❆ss✐♠✱ C2 H = F D ❡ C2 F = HD✳ ▲♦❣♦✱ ♦ q✉❛❞r✐❧át❡r♦ C2 F DH é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❈♦♠♦ a + b = π/2✱ ♦ â♥❣✉❧♦ ∠F C2 H = ∠F DH = π/2✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ▲♦❣♦✱ ♦ â♥❣✉❧♦ ❡♥tã♦ ❡♠ q✉❡stã♦ é ✉♠ r❡tâ♥❣✉❧♦✳ Pr♦♣♦s✐çã♦ ✸✳✸✳ ❖ s❡❣♠❡♥t♦ C2 D é ❝♦♥❣r✉❡♥t❡ ❛♦ s❡❣♠❡♥t♦ FH ❡ ✐♥t❡rs❡❝t❛♠✲s❡ ❡♠ s❡✉s ♣♦♥t♦s ♠é❞✐♦s✳ ❉❡♠♦♥str❛çã♦✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✭✸✳✷✮✱ ♦ q✉❛❞r✐❧át❡r♦ C2 F DH é ✉♠ r❡tâ♥❣✉❧♦✳ ❯♠❛ ✈❡③ q✉❡ ❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ sã♦ ❝♦♥❣r✉❡♥t❡s ❡ ✐♥t❡rs❡❝t❛♠ ❡♠ s❡✉s ♣♦♥t♦s ♠é❞✐♦s✱ s❡❣✉❡ q✉❡ ♦s s❡❣♠❡♥t♦s C2 D ❡ FH sã♦ ❝♦♥❣r✉❡♥t❡s ❡ s❡ ✐♥t❡rs❡❝t❛♠ ❡♠ s❡✉s ♣♦♥t♦s ♠é❞✐♦s✳ Pr♦♣♦s✐çã♦ ✸✳✹✳ ❖ s❡❣♠❡♥t♦ FH é t❛♥❣❡♥t❡ ❝♦♠✉♠ ❛ ❛♠❜♦s ♦s s❡♠✐❝ír❝✉❧♦s ♠❡♥♦r❡s✳ ❉❡♠♦♥str❛çã♦✳ P❡❧❛ ♣r♦♣♦s✐❝ã♦ ✭✸✳✷✮✱ s❛❜❡♠♦s q✉❡ s✐çã♦ ✭✸✳✸✮✱ ✈✐♠♦s q✉❡ ♦s s❡❣♠❡♥t♦s FH N F = N C2 = N H ✳ ❙❡❣✉❡ ❞✐ss♦ q✉❡ é ✉♠ r❡tâ♥❣✉❧♦ ❡ ♣❡❧❛ Pr♦♣♦✲ C2 D sã♦ ❝♦♥❣r✉❡♥t❡s ❡ ✐♥t❡rs❡❝t❛♠✲s❡ ♥♦s s❡✉s N ❛ ✐♥t❡rs❡❝çã♦ ❞❡ss❡s s❡❣♠❡♥t♦s t❡♠♦s ❡♥tã♦ ♦s tr✐â♥❣✉❧♦s △N C2 F ❡ △N C2 H sã♦ tr✐â♥❣✉❧♦s ❡ r❡s♣❡❝t✐✈♦s ♣♦♥t♦s ♠é❞✐♦s✳ ❈❤❛♠❛♥❞♦ ❞❡ q✉❡ C2 F DH ✹✷ ❈❆P❮❚❯▲❖ ✸✳ ❆❘❇❊▲❖❙ ∠N C2 H = ∠N HC2 = a✱ ∠N C2 F = ∠N F C2 = b✳ ❙❡❥❛♠ I ❡ J ♦s ♣♦♥t♦s ♠é❞✐♦s ❞♦s s❡❣♠❡♥t♦s C1 C2 ❡ C2 C3 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❈♦♥str✉í♠♦s ❡♥tã♦ ♦s s❡❣♠❡♥t♦s IF ❡ JH ✳ ❚❡♠♦s q✉❡ IF = IC2 ✱ JH = JC2 ❡ ♦s tr✐â♥❣✉❧♦s △F IC2 ❡ △HC2 J sã♦ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s✳ P♦rt❛♥t♦✱ ♦ â♥❣✉❧♦ ✐sós❝❡❧❡s✳ ❆ss✐♠✱ ∠HF I = ∠IF C2 + ∠C2 F N ∠HF I = a + b ∠HF I = π/2, ❡ ♦ â♥❣✉❧♦ ∠F HJ = 90◦ ✱ ♣♦✐s✿ ∠F HJ = ∠N HC2 + ∠C2 HJ ∠F HJ = a + b ∠F HJ = π/2. ❈♦♠♦ IF ❛♦ s❡❣♠❡♥t♦ é ✉♠ r❛✐♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ FH ❡♥tã♦ ♦ s❡❣♠❡♥t♦ ♠❡s♠♦ ♠♦❞♦✱ ♦ s❡❣♠❡♥t♦ FH JH ❆ss✐♠✱ ♦ s❡❣♠❡♥t♦ FH C1 C2 ❡ ♦ s❡❣♠❡♥t♦ IF JC2 ✳ é ♣❡r♣❡♥❞✐❝✉❧❛r é t❛♥❣❡♥t❡ ❛♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ é ✉♠ r❛✐♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ é ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ s❡❣♠❡♥t♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ FH P♦rt❛♥t♦✱ ♦ s❡❣♠❡♥t♦ FH C2 C3 ✱ C1 C2 ✳ ❉♦ ♦ s❡❣♠❡♥t♦ t❛♠❜é♠ é t❛♥❣❡♥t❡ ❛♦ C2 C3 ✳ é ❛ t❛♥❣❡♥t❡ ❝♦♠✉♠ ❛ ❛♠❜♦s ♦s s❡♠✐❝ír❝✉❧♦s ♠❡♥♦r❡s✳ ❆♥t❡s ❞❡ ♠♦str❛r♠♦s ❛ ♣ró①✐♠❛ ♣r♦♣r✐❡❞❛❞❡✱ ❝❛❧❝✉❧❛♠♦s ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ ✭❞✐❛❣♦♥❛❧✮✳ ❉❡♥♦t❛♥❞♦✲s❡ C1 C2 = r ✱ t❡♠♦s q✉❡✱ C2 C3 = 1 − r ✱ C2 D ❧♦❣♦ C2 D 2 = C1 C2 · C2 C3 C2 D2 = r(1 − r) p C2 D = r(1 − r) √ r − r2 C2 D = Pr♦♣♦s✐çã♦ ✸✳✺✳ ❆ ár❡❛ ❞❡ ✉♠ ❛r❜❡❧♦ é ✐❣✉❛❧ à ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ ❉❡♠♦♥str❛çã♦✳ ❞✐â♠❡tr♦ ❡ C2 C3 ✱ ▼❛s✱ C1 C3 ❆ ár❡❛ A ❞♦ ❛r❜❡❧♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣❡❧❛ ár❡❛ A2 ❡ A3 ❞♦s A = A1 − (A2 + A3 )✳ s✉❜tr❛í❞❛ ❞❛ s♦♠❛ ❞❛s ár❡❛s r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❖✉ s❡❥❛✱ A1 C2 D ✳ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ s❡♠✐❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s C1 C2 ✸✳✶✳ ✹✸ ❆❘❇❊▲❖❙ ●✃▼❊❖❙ π· A1 = π· 2  r 2 = π , 8 πr2 , 8 2 2 π · 1−r π · 1−2r+r π − 2πr + πr2 2 4 = = . = 2 2 8 A2 = A3  1 2 2 2 2 = ▲♦❣♦ ❛ ár❡❛ ❞♦ ❛r❜❡❧♦ é✿   2 π π − 2πr + πr2 πr A = , − + 8 8 8 π πr2 π 2πr πr2 − − + − , A = 8 8 8 8 8 πr2 2πr πr2 + − , A = − 8 8 8 2πr 2πr2 A = − , 8 8 2πr − 2πr2 A = , 8 πr − πr2 . A = 4 ❆❣♦r❛✱ ❝❛❧❝✉❧❛♠♦s ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C2 D✿ √ r − r2 A = π· 2 2 r−r , A = π· 4 πr − πr2 , A = 4 2 , ♦ q✉❡ ♠♦str❛ q✉❡ ❛ ár❡❛ ❞♦ ❛r❜❡❧♦ é ✐❣✉❛❧ à ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C2 D✳ ✸✳✶ ❆r❜❡❧♦s ❣ê♠❡♦s ❆ ✜❣✉r❛ ❛ s❡❣✉✐r ♠♦str❛ ♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ♥♦ ❛r❜❡❧♦✳ ❖s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s sã♦ ❝ír❝✉❧♦s ✐♥t❡r✐♦r❡s ♥♦ ❛r❜❡❧♦ q✉❡ sã♦ t❛♥❣❡♥t❡s ❛♦ s❡❣♠❡♥t♦ ♣❡r♣❡♥❞✐❝✉❧❛r C2 D ❛♦ s❡❣♠❡♥t♦ C1 C3 ♥♦ ♣♦♥t♦ C2 ✱ q✉❡ é ❛ ✐♥t❡rs❡❝çã♦ ❞♦s ❞♦✐s s❡♠✐❝ír❝✉❧♦s ♠❡♥♦r❡s✳ ✹✹ ❈❆P❮❚❯▲❖ ✸✳ ❆❘❇❊▲❖❙ ❋✐❣✉r❛ ✸✳✸✿ ❈ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s Pr♦♣♦s✐çã♦ ✸✳✻✳ ❖s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s tê♠ ♦ ♠❡s♠♦ r❛✐♦✳ ❋✐❣✉r❛ ✸✳✹✿ ❈ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s C1 C3 = 1✱ C1 C2 = r✱ M ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ C1 C3 ✱ P ♦ ❝❡♥tr♦ ❞♦ ←−→ ❝ír❝✉❧♦ ❣ê♠❡♦ ❞❛ ❞✐r❡✐t❛✱ J ❛ ♣r♦❥❡çã♦ ♦rt♦❣♦♥❛❧ ❞❡ P s♦❜r❡ ❛ r❡t❛ C1 C3 ✱ F ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ C2 C3 ❡ R ♦ r❛✐♦ ❞❡st❡ ❝ír❝✉❧♦ ✭❝ír❝✉❧♦ ❞❛ ❞✐r❡✐t❛✮✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ ✸✳✶✳ ✹✺ ❆❘❇❊▲❖❙ ●✃▼❊❖❙ ❚❡♠♦s q✉❡✿ MP = 1 − R, 2 1 M J = M C3 − JF − F C3 = − 2     1 1−r 1−r =R+r− . −R − 2 2 2 ▲♦❣♦✿ JP 2 = M P 2 − M J 2 =  2  2   1 1 1 1 2 2 2 = −R − R+r− = − R + R − R + r + + 2Rr − R − r 2 2 4 4 ❊♥tã♦ t❡♠♦s q✉❡✿ JP 2 = r − r2 − 2Rr. ✭✸✳✶✮ ❆❧é♠ ❞✐st♦✱ 1−r + R, 2 1−r JF = − R. 2 FP = ▲♦❣♦✱ JP 2 2 2 = F P − JF =  1 r − +R 2 2 2 2 1 r − −R = 2 2  r2 r 2 + + R − − R + Rr 4 2  −  r 1 1 r2 2 + + R − + R − Rr − = 4 4 2 4 ❆ss✐♠✱ JP 2 = 2R − 2Rr. ✭✸✳✷✮ ■❣✉❛❧❛♥❞♦✲s❡ ❛s ❡q✉❛çõ❡s ✭✸✳✶✮ ❡ ✭✸✳✷✮✱ t❡♠♦s✿ r − r2 − 2Rr = 2R − 2Rr. ▲♦❣♦✱ R= ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ ♦ r❛✐♦ r(1 − r) ✳ ❙❡ I ❞❡♥♦t❛ ❛ ❡sq✉❡r❞❛ é 2 ♣♦♥t♦ ♠é❞✐♦ ❞❡ C1 C2 ✳ ❚❡♠♦s q✉❡✿ ♣r♦❥❡çã♦ ❞♦ ❝❡♥tr♦ r + R1 , 2 r EI = − R1 . 2 EP1 = r(1 − r) . 2 R1 ❞♦ ❝ír❝✉❧♦ ❞❡ ❆rq✉✐♠❡❞❡s ❞❛ P1 ❞❡st❡ ❝ír❝✉❧♦ s♦❜r❡ ←−→ C1 C3 ✱ E ♦ ✹✻ ❈❆P❮❚❯▲❖ ✸✳ ▲♦❣♦✿ IP12 = EP12 − EI 2 = ❆❧é♠ ❞✐st♦ M P1 = r 2 + R1 1 − R1 2 IM = C1 M − C1 E − EI = 2 − r 2 − R1 2 = 2rR1 ❆❘❇❊▲❖❙ ✭✸✳✸✮  1 1 r r − − − R1 = − r − R1 2 2 2 2 ▲♦❣♦✿ IP12 =  1 −R 2 2 −  1 − r + R1 2 2 = 2rR1 − 2R1 − r2 + r ✭✸✳✹✮ ■❣✉❛❧❛♥❞♦✲s❡ ❛s ❡q✉❛çõ❡s ✭✸✳✸✮ ❡ ✭✸✳✹✮✱ t❡♠♦s✿ 2rR1 − 2R1 − r2 + r = 2rR1 . ■st♦ é R1 = R R1 = ▲♦❣♦✱ r(1 − r) . 2 ❝♦♠♦ s❡ tr❛t❛✈❛ ♣r♦✈❛r✳ ❱❡r❡♠♦s ❛❣♦r❛ ❝♦♠♦ ❡♥❝♦♥tr❛r ♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ♣♦r ♠❡✐♦ ❞❡ s✉❛ ❝♦♥str✉çã♦✳ EN ❡ F G ♠❡❞✐❛tr✐③❡s ❞♦s s❡❣♠❡♥t♦s C1 C2 ❡ C2 C3 ✱ ❝♦♠ E ❡ F ♦s ♣♦♥t♦s ❡ C2 C3 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ N é ❛ ✐♥t❡rs❡❝çã♦ ❞❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ C1 C2 ❝♦♠ ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C1 C2 ❡ G é ❛ ✐♥t❡rs❡❝çã♦ ❞❛ ♠❡❞✐❛tr✐③ ❞♦ s❡❣♠❡♥t♦ C2 C3 ❝♦♠ ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ C2 C3 ✳ P❡❧♦ ♣♦♥t♦ N ❝♦♥str✉í♠♦s ♦ s❡❣♠❡♥t♦ N F q✉❡ ✐♥t❡rs❡❝t❛ C2 D ♥♦ ♣♦♥t♦ H ✳ ❈♦♥str✉í♠♦s ♠é❞✐♦s ❞❡ C1 C2 P❡❧♦ ❝❛s♦ ❞❡ s❡♠❡❧❤❛♥ç❛ ❆❆✱ ♦s tr✐â♥❣✉❧♦s C2 H r/2 r 2 C2 H = △EN F ❡ △C2 HF sã♦ s❡♠❡❧❤❛♥t❡s✳ ▲♦❣♦✱ (1 − r)/2 1/2 1 1−r r · (1 − r). C2 H = 2 = ❊♠ s❡❣✉✐❞❛✱ ❝♦♥str✉í♠♦s ♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ ❡♠ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r C1 C3 ♥♦s ♣♦♥t♦s I ❡ C2 ❡ r❛✐♦ C2 H ❞❡ ♠♦❞♦ ❛ ✐♥t❡rs❡❝t❛r ❛ J✳ ❈♦♥str✉í♠♦s ❛s r❡t❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❛ C1 C3 ♣❛ss❛♥❞♦ ♣♦r I ❡ J ✳ ✸✳✶✳ ❆❘❇❊▲❖❙ ●✃▼❊❖❙ ✹✼ ❋✐❣✉r❛ ✸✳✺✿ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ❋✐❣✉r❛ ✸✳✻✿ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ❉❛í ❝♦♥str✉í♠♦s ♦s ❝ír❝✉❧♦s ❝❡♥tr❛❞♦s ❡♠ F ❡ E ❝♦♠ r❡s♣❡❝t✐✈♦s r❛✐♦s FI ❡ EJ ✳ ❆s L ❡ P ✳ ❙❡♥❞♦ ♦ ♣♦♥t♦ L1 ❛ ✐♥t❡rs❡❝çã♦ ❞♦ s❡❣♠❡♥t♦ LE ❝♦♠ ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ r❛✐♦ EC2 ❡ P1 ❛ ✐♥t❡rs❡çã♦ ❞❡ P F ❝♦♠ ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ r❛✐♦ F C2 ✱ t❡♠♦s ❛❣♦r❛ q✉❡ ♦s s❡❣♠❡♥t♦s LL1 ❡ P P1 sã♦ ♦s r❛✐♦s ❞♦r ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s✳ ✐♥t❡rs❡❝çõ❡s ❞❛s ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❝♦♠ ❡st❡s ❝ír❝✉❧♦s sã♦ ♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s✱ ❆s ❝♦♦r❞❡♥❛❞❛s ❞♦s ❝❡♥tr♦s ❞❡ss❡s ❛r❜❡❧♦s sã♦ ❡①♣r❡ss❛s ♣♦r✿ L = (x1 , y1 ) = r 2  √ (1 + r), r 1 − r ❡ P = (x2 , y2 ) = r 2 √  (3 − r), (1 − r) r ✱ ♣♦✐s ❝♦♠♦ ✹✽ ❈❆P❮❚❯▲❖ ✸✳ ❋✐❣✉r❛ ✸✳✼✿ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ❋✐❣✉r❛ ✸✳✽✿ ❈♦♥str✉çã♦ ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s r R = HC2 = (1 − r)✱ 2 t❡♠♦s✿ r r x1 = r − (1 − r) = (1 + r), 2 2 ❆❘❇❊▲❖❙ ✸✳✶✳ ❆❘❇❊▲❖❙ ●✃▼❊❖❙ ✹✾ ❆❧é♠ ❞✐ss♦✱ IL2 = EL2 − EI 2 , ♠❛s✱   r − r2 r2 r + ❡ =r− EL = 2 2 2   r2 r − r2 r = EI = EC2 − IC2 = − 2 2 2    2 2 2 2 r r y12 = r− − , 2 2 r4 r4 − , y12 = r2 − r3 + 4 4 y12 = r2 (1 − r), p y1 = r2 (1 − r), √ y1 = r 1 − r. ❡♥tã♦✱ ❡ r r2 3r − r2 r(3 − r) r = = , x2 = r + (1 − r) = r + − 2 2 2 2 2 ❡ JP 2 = F P 2 − JF 2 , ♠❛s✱     1−r 1 r2 r − r2 JF = − = −r+ ❡✱ 2 2 2 2     1 r2 r − r2 1−r + = − ❡♥tã♦✱ FP = 2 2 2 2 2  2  r2 1 1 r2 2 − , − −r+ y2 = 2 2 2 2 r4 r2 1 r2 r4 1 2 2 − + − −r − +r− + r3 , y2 = 4 4 4 4 4 2 y22 = −2r2 + r + r3 , y22 = r(r2 − 2r + 1), y22 = r(1 − r)2 , p y2 = r(1 − r)2 , √ y2 = (1 − r) r. ✺✵ ❈❆P❮❚❯▲❖ ✸✳ ❆❘❇❊▲❖❙ ❋✐❣✉r❛ ✸✳✾✿ ❈♦♦r❞❡♥❛❞❛s ❞♦s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s ✸✳✷ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s ◆♦ ❧✐✈r♦ ■❱ ❞❡ s✉❛ ❝♦❧❡çã♦✱ P❛♣♣✉s ❡st❡♥❞❡ ♦ ❡st✉❞♦ q✉❡ ❆rq✉✐♠❡❞❡s ❢❡③ s♦❜r❡ ♦s ❛r❜❡❧♦s✳ ◆❛ ✜❣✉r❛ s❡❣✉✐♥t❡ ✈❡♠♦s ♦s ❝ír❝✉❧♦s t❛♥❣❡♥t❡s ❛♦s s❡♠✐❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s C1 C3 ✱ C1 C2 ❡ C2 C3 ✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ❡st❡s ❝ír❝✉❧♦s é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ❋✐❣✉r❛ ✸✳✶✵✿ ❈❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ❱❛♠♦s ❛❣♦r❛ ♠♦str❛r q✉❡ ♦s ❝❡♥tr♦s ❞❡st❡s ❝ír❝✉❧♦s ❞❡s❝r❡✈❡♠ ✉♠❛ ❡❧✐♣s❡ ❞❡ ❢♦❝♦s q✉❡ ❡stã♦ ❝♦♥t✐❞♦s ♥❛ r❡t❛ q✉❡ ♣❛ss❛ ♣♦r C1 C3 ✳ ✸✳✷✳ ✺✶ ❈❆❉❊■❆ ❉❊ P❆PP❯❙ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ❛♣❡♥❛s ♦ s❡♠✐❝ír❝✉❧♦ ♠❛✐♦r ❞❡ ❝❡♥tr♦ ❡ ✉♠ s❡♠✐❝ír❝✉❧♦ ♠❡♥♦r✱ ❞❡ H ❞❡ ✉♠ ❝ír❝✉❧♦ t❛♥❣❡♥t❡ ❛ ❛♠❜♦s ♣❡rt❡♥❝❡ ❛ ✉♠❛ ❡❧✐♣s❡ ❞❡ ❢♦❝♦s D ❡ E ✳ ❉❡ ❢❛t♦ s❡❥❛♠ G ❡ I ♦s ♣♦♥t♦s ❞♦s s❡♠✐❝➥❝✉❧♦s ❞❡ ❝❡♥tr♦s D ❡ E ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s q✉❡ H ❡stá ❡♥tr❡ D ❡ G✱ ❡ I ❡stá ❡♥tr❡ E ❡ H ✳ ❝❡♥tr♦ E✱ D ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ ♦ ❝❡♥tr♦ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ DC1 EC1 F C2 HI HE HD 2R HE + HD P♦rt❛♥t♦ ♦ ♣♦♥t♦ ❡ F H = = = = = = = = DC3 = DG = R, EC2 = EI = r1 , F C 3 = r2 , HG = r, r1 + r, R − r, 2r1 + 2r2 , R + r1 = r1 + r2 + r1 = 2r1 + r2 = C1 F. ♣❡rt❡♥❝❡ ❛ ✉♠❛ ❡❧✐♣s❡ ❞❡ ❢♦❝♦s ✭♣♦♥t♦ ♠é❞✐♦ ❞❡ E ❡ D ❝✉❥♦s ✈ért✐❝❡s sã♦ ♦s ♣♦♥t♦s C2 C3 ) ✳ ❋✐❣✉r❛ ✸✳✶✶✿ ❖s ❝❡♥tr♦s ❞♦s ❝ír❝✉❧♦s ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s✳ ❱❡r❡♠♦s ❛ s❡❣✉✐r ❝♦♠♦ ❝♦♥str✉✐r ❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ♥♦s ❛r❜❡❧♦s✳ C1 ✺✷ ❈❆P❮❚❯▲❖ ✸✳ • ❙♦❜r❡ ✉♠❛ r❡t❛ r ✱ t♦♠❛♠♦s ✸ ♣♦♥t♦s ❞✐st✐♥t♦s ❡ ❝♦❧✐♥❡❛r❡s C1 ❞❛ C3 ✱ ❡ ♦s s❡♠✐❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s C1 C2 , C1 C3 r❡t❛ r ✳ ❡ • ❉❡♣♦✐s ❝♦♥str✉í♠♦s ✉♠ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ • ❆ ✐♥✈❡rsã♦ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ ❛ r ❡ q✉❡ ♣❛ss❛ ♣♦r ❛ r❡t❛ t ′ ❞✐â♠❡tr♦ C2 ✮ C1 C2 I ❡ C 1 , C2 C2 C3 ❞❡ ❝❡♥tr♦ ❡♠ r❡❧❛çã♦ ❛ C3 ✱ ❝♦♠ C2 ❡♥tr❡ t♦❞♦s ❞❡ ✉♠ ♠❡s♠♦ ❧❛❞♦ C1 I ❡ ❆❘❇❊▲❖❙ ❡ r❛✐♦ C1 C3 ✳ é ❛ r❡t❛ ❡ ❛ ✐♥✈❡rsã♦ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ s′ C1 C3 ✭♣❡r♣❡♥❞✐❝✉❧❛r ❡♠ r❡❧❛çã♦ ❛ I é ✭✉♠❛ ✈❡③ q✉❡ ♣❛ss❛♠ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✱ ❡ ❛ ✐♥✈❡rsã♦ ❞♦ ❝ír❝✉❧♦ ❞❡ C2 C3 O′ é ♦ ❝ír❝✉❧♦ K0′ ❞❡ ❝❡♥tr♦ O′ ✳ • P❡❧♦ ♣♦♥t♦ • P❛r❛ ❝♦♥str✉✐r ♦ ♣ró①✐♠♦ ❝ír❝✉❧♦ ✐♥✈❡rt❡♠♦s ♦ ❝ír❝✉❧♦ ❝♦♥str✉í♠♦s ✉♠❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛ C1 C3 ❡♠ A✳ ❡♠ r❡❧❛çã♦ ❛♦ ♣♦♥t♦ A q✉❡ ✐♥t❡rs❡❝t❛ K0′ K0′ ♦❜t❡♥❞♦✲s❡ ❛ss✐♠ ♦ ❝ír❝✉❧♦ K1′ ✳ • P❛r❛ ❝♦♥str✉✐r ♦s ❞❡♠❛✐s ❝ír❝✉❧♦s s❡❣✉✐♠♦s ♣r♦❝❡❞✐♠❡♥t♦ ❛♥t❡r✐♦r✳ • ❆❣♦r❛ ♣❛r❛ ❝♦♥str✉✐r ♦s ❝ír❝✉❧♦s ❞❛ ❝❛❞❡✐❛ ❜❛st❛ ✐♥✈❡rt❡r♠♦s ♦s ❝ír❝✉❧♦s K3′ ❡♠ r❡❧❛çã♦ ❛ I ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❝ír❝✉❧♦s K1 ✱ K2 ✱ ✳✳✳ ✱ Kn ✳ ❋✐❣✉r❛ ✸✳✶✷✿ ❈♦♥str✉çã♦ ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ✉s❛♥❞♦ ✐♥✈❡rsã♦✳ K1′ ✱ K2′ ✱ ✳✳✳✱ ✸✳✷✳ ✺✸ ❈❆❉❊■❆ ❉❊ P❆PP❯❙ ❖❜s❡r✈❛çã♦ ✸✳✶✳ ❆ ♣r♦♣r✐❡❞❛❞❡ ❞❛ t❛♥❣ê♥❝✐❛ é ♣r❡s❡r✈❛❞❛ ♣♦r ✐♥✈❡rsã♦✳ ❚❡♦r❡♠❛ ✸✳✶✳ ❙❡❥❛♠ C 1 , C2 ❡ C3 três ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s t❛✐s q✉❡ C2 ❡stá ❡♥tr❡ C1 ❡ C3 C1 C2 , C1 C3 ❡ C2 C3 ✳ ❙❡❥❛♠ O0 , O1 , . . . , On ♦s ❝❡♥tr♦s ❡ s❡❥❛♠ ♦s ❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s ❞♦s ❝ír❝✉❧♦s ❝❤❛♠❛r♠♦s K0 , K1 , . . . , Kn ❞❛ ❈❛❞❡✐❛ ❞❡ rn ♦ r❛✐♦ ❞❡ Kn ❡ hn ❛ ❞❡ P❛♣♣✉s✱ ❝♦♥str✉í❞❛ ♥❛ Pr♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ❙❡ ❞✐stâ♥❝✐❛ ❞❡ On ❛ ←−→ C1 C3 ✱ ❡♥tã♦ hn = 2nrn ✳ ❋✐❣✉r❛ ✸✳✶✸✿ ❉✐stâ♥❝✐❛ ❞♦ ❝❡♥tr♦ ❞❛ ❝❛❞❡✐❛ ❞❡ P❛♣♣✉s ❛té ❛ r❡t❛ s✉♣♦rt❡✳ ✶✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ✐♥✈❡rsã♦ r❡❧❛t✐✈❛ ❛♦ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ C1 ❡ r❛✐♦ On Tn ✱ ❡♠ q✉❡ Tn é ♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❛♦ s❡♠✐❝ír❝✉❧♦ Kn ❞❛ s❡♠✐r❡t❛ ❞❡ ♦r✐❣❡♠ C1 ✳ ❉❡♠♦♥str❛çã♦✳ ✷✳ ❋❛③❡♠♦s ✐ss♦✱ ♣♦rq✉❡ C1 Tn é ♣❡r♣❡♥❞✐❝✉❧❛r ❛ On Tn ✱ s❡♥❞♦ C1 Tn t❛♥❣❡♥t❡ ❛ Kn ❡ On Tn t❛♥❣❡♥t❡ ❛♦ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦ ❡✱ ♣♦r ✐ss♦✱ Kn é ♦rt♦❣♦♥❛❧ ❛♦ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦✳ ❖s ❝ír❝✉❧♦s ♦rt♦❣♦♥❛✐s ❛♦ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡sã♦ sã♦ ✐♥✈❡rs♦s ❞❡ s✐ ♠❡s♠♦s✳ ✸✳ ❖s ✐♥✈❡rs♦s ❞♦s ❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s C1 C2 ❡ C1 C3 sã♦ r❡t❛s ✱ ♣♦✐s ♦s ❝ír❝✉❧♦s ♣❛ss❛♠ ♣♦r C1 ✱ q✉❡ é ❝❡♥tr♦ ❞❛ ✐♥✈❡rsã♦✳ ❊ss❛s r❡t❛s ✜❝❛♠ ❞❡✜♥✐❞❛s ♣❡❧♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ss❡s ❝ír❝✉❧♦s ❝♦♠ ♦ ❝ír❝✉❧♦ ❞❡ ✐♥✈❡rsã♦✳ ✹✳ ❖ ✐♥✈❡rs♦ ❞♦ ❝ír❝✉❧♦ Kn ❞❡ ❝❡♥tr♦ On é ✉♠ ❝ír❝✉❧♦ t❛♥❣❡♥t❡ às ❞✉❛s r❡t❛s ✭❞❡t❡r♠✐✲ ♥❛❞❛s ❝♦♠♦ ✐♥✈❡rs♦s ❞♦s ❝ír❝✉❧♦s ❞❡ ❞✐â♠❡tr♦s C1 C2 ❡ C1 C3 ✮ ❞❡ ❝❡♥tr♦ Pn ✳ ✺✳ ❖s ✐♥✈❡rs♦s Ki′ ✱ 0 ≤ i ≤ n sã♦ t❛♥❣❡♥t❡s ❛ ❡st❛s r❡t❛s ❡ tê♠ ♦ ♠❡s♠♦ r❛✐♦ On Tn ✳ ✺✹ ❈❆P❮❚❯▲❖ ✸✳ ✻✳ O n Pn é ✐❣✉❛❧ à s♦♠❛ ❞❡ ✉♠ r❛✐♦ ❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦s ❞✐â♠❡tr♦s ❞❡ 2rn + 2(n − 1)rn = 2nrn ✳ ❆❘❇❊▲❖❙ K0′ ❡ ♦✉tr♦ ♣❛r❛ Kn′ = Kn ❛❧é♠ ❞❡ 2(n − 1)rn n − 1 ❝ír❝✉❧♦s ✐❣✉❛✐s ❛ Kn ✱ Kn′ ✱ 1 ≤ i ≤ n − 1✿ ✸✳✸ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛ ❊♠ ♠❡✐♦ ❛♦s ❝ír❝✉❧♦s ✐♥❞✐✈✐❞✉❛✐s ❞❡ ❆rq✉✐♠❡❞❡s✱ ❤á ✉♠ ♦✉tr♦ ❝ír❝✉❧♦ ❞❡♥tr♦ ❞♦s ❛r❜❡❧♦s q✉❡ é ❝♦♥❣r✉❡♥t❡ ❛♦s ❝ír❝✉❧♦s ✐♥❞✐✈✐❞✉❛✐s✳ ➱ ♦ ❝❤❛♠❛❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ▲❡♦♥ ❇❛♥❦♦✛✱ ❞❡♥t✐st❛ ❡♠ ❇❡✈❡r❧② ❍✐❧❧s ❡ t❛♠❜é♠ ✉♠ ♠❛t❡♠át✐❝♦✱ ❡♥❝♦♥tr♦✉✲♦ ❡ ❡s❝r❡✈❡✉ ✉♠ ❛rt✐❣♦ ❝❤❛♠❛❞♦ ✏❖s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s sã♦ ♠❡s♠♦s ❣ê♠❡♦s❄✑ ❖ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛ é ❢♦r♠❛❞♦ ❛ ♣❛rt✐r ❞♦ ❛r❜❡❧♦✳ ✉♠ ❝ír❝✉❧♦ ❞❡ P❛♣♣✉s✳ K1 Pr✐♠❡✐r❛♠❡♥t❡✱ ❝♦♥str✉í♠♦s t❛♥❣❡♥t❡ ❛ ❝❛❞❛ s❡♠✐❝ír❝✉❧♦ ❞♦ ❛r❜❡❧♦✱ ❝♦♠♦ ♥❛ ❝♦♥str✉çã♦ ❞❛ ❈❛❞❡✐❛ ❖ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛ é ♦ ❝ír❝✉❧♦ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ ❞❡ t❛♥❣ê♥❝✐❛ ❞♦s s❡♠✐❝ír❝✉❧♦s ♠❡♥♦r❡s ❞♦ ❛r❜❡❧♦✱ ❡ ♣❡❧♦s ♣♦♥t♦s ❞❡ t❛♥❣ê♥❝✐❛ ❞❡ K1 ❝♦♠ ♦s s❡♠✐❝ír❝✉❧♦s ♠❡♥♦r❡s ❞♦ ❛r❜❡❧♦✳ ❋✐❣✉r❛ ✸✳✶✹✿ ❈ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ ❖ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛ ❡ ♦s ❝ír❝✉❧♦s ❣ê♠❡♦s ❞❡ ❆rq✉✐♠❡❞❡s sã♦ ✐❞ê♥t✐❝♦s✱ r ♦✉ s❡❥❛✱ s❡✉ r❛✐♦ é .(1 − r)✳ Pr♦♣♦s✐çã♦ ✸✳✼✳ 2 ✸✳✸✳ ✺✺ ❈❮❘❈❯▲❖ ❉❊ ❇❆◆❑❖❋❋ ❈♦♥s✐❞❡r❡♠♦s C1 C3 = 1 ✱ C1 C2 = r ❡ x ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✳ Pr✐♠❡✐r♦✱ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △ABC r❡♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ✭✸✳✶✹✮✱ q✉❡ ✈❛♠♦s ✐♥❞✐❝❛r ♣♦r A✳ ❚❡♠♦s ❡♥tã♦ q✉❡✿ ❉❡♠♦♥str❛çã♦✳ 1 , 2 r + R, AC = 2 1−r BC = + R. 2 AB = ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r♦♣♦s✐çã♦ ❞❡ P❛♣♣✉s✱ ❛ ❛❧t✉r❛ ❞❡ C r❡❧❛t✐✈❛ ❛ ❜❛s❡ AB é ❞✉❛s ✈❡③❡s ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ K ✱ ♦✉ s❡❥❛✱ ❛ ❛❧t✉r❛ h s❡rá ✐❣✉❛❧ ❛ 2R✳ ❈❛❧❝✉❧❛♥❞♦✲s❡ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △ABC t❡♠♦s✿ 1 · 2R R A= 2 = . 2 2 ✭✸✳✺✮ ▲♦❣♦✱ ❞❡ ❛❝♦r❞♦ p ❝♦♠ ❛ ❢ór♠✉❧❛ ❞❡ ❍❡r♦♥✱ s❛❜❡♠♦s q✉❡ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❞❡ ❧❛❞♦s a✱ b ❡ c é A = s · (s − a) · (s − b) · (s − c)✱ ♦♥❞❡ s é ♦ s❡♠✐♣❡rí♠❡tr♦ ❞♦ tr✐â♥❣✉❧♦✳ ❙❡❥❛ ❛❣♦r❛✿ 1−r + R, 2 r b = AC = + R, 2 1 c = AB = . 2 a = BC = ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s = s−a = s−b = s−c = 1 r 1−r + +R+ a+b+c 2 = 1 + R, = 2 2 2 2 2 1 1−r r +R− +R= , 2 2 2 r 1−r 1 +R− +R= , 2 2 2 1 1 + R − = R. 2 2 ❊♥tã♦ ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ❍❡r♦♥ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ △ABC s❡rá ❞❛❞❛ ♣♦r✿ A= s s       r 1 1−r R +R · · · (R) = · r · (1 − r) · (1 + 2R) 2 2 2 8 ✭✸✳✻✮ ✺✻ ❈❆P❮❚❯▲❖ ✸✳ ❆❘❇❊▲❖❙ ❙❛❜❡♠♦s ❛✐♥❞❛ q✉❡ ❛ ár❡❛ A ❞♦ tr✐â♥❣✉❧♦ △ABC é ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ s♦♠❛♥❞♦✲s❡ ❛s ár❡❛s ❞♦s tr✐â♥❣✉❧♦s △AOB ✱ △AOC ❡ △BOC ✱ q✉❡ ❞❡s✐❣♥❛r❡♠♦s ♣♦r A1 , A2 ❡ A3 r❡s♣❡❝t✐✈❛♠❡♥t❡ ❡✱ ❧❡♠❜r❛♥❞♦ q✉❡ x é ♦ r❛✐♦ ❞♦ ❝ír❝✉❧♦ ❞❡ ❇❛♥❦♦✛✱ t❡♠♦s q✉❡✿ A = A1 + A2 + A3 ,    1 r 1 1−r 1 1 · ·x+ · +R ·x+ · + R · x, A = 2 2 2 2 2 2   1 1 r 1 r A = ·x· + +R+ − +R . 2 2 2 2 2 ▲♦❣♦✱ A= x · (1 + 2R) 2 ✭✸✳✼✮ A2 P♦r ✭✸✳✺✮✱ ✭✸✳✻✮ ❡ ✭✸✳✼✮✱ ❡ ✉s❛♥❞♦✲s❡ q✉❡ A = ✱ ❡♥tã♦ A R = 2 R = 2 R = 2 2 = x = s R · r · (1 − r) · (1 + 2R) 8 x · (1 + 2r) 2 R · r · (1 − r) · (1 + 2R) 8 , x · (1 + 2R) 2 R · r · (1 − r) 4 , x r(1 − r) , x r(1 − r) . 2 ! 2 , ❈❛♣ít✉❧♦ ✹ ❆♣❧✐❝❛çõ❡s ◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❡①❡♠♣❧♦s ❞❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s✱ q✉❡ ♣♦❞❡♠ s❡r ❢❡✐t❛s ✉t✐❧✐③❛♥❞♦✲s❡ ❣❡♦♠❡tr✐❛ ❞✐♥â♠✐❝❛✱ ♥♦ ❝❛s♦✱ ♦ ❣❡♦❣❡❜r❛✳ ❚❛✐s ❝♦♥str✉çõ❡s t❛♠❜é♠ ♣♦❞❡rã♦ s❡r r❡❛❧✐③❛❞❛s ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✳ ✹✳✶ ❊①❡♠♣❧♦s ❊①❡♠♣❧♦ ✹✳✶✳ ❈♦♥str✉çã♦ ✶✳ • ❈♦♥str✉❛ ✉♠❛ r❡t❛ r❀ • ▼❛rq✉❡ ♦s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s A✱ B ❡ C s♦❜r❡ r✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ ♣♦♥t♦ B ✜q✉❡ 1 2 ❡♥tr❡ ♦s ♣♦♥t♦s A ❡ C ❡ q✉❡ AC = 1✱ AB = ❡ BC = ❀ 3 3 • ❈♦♥str✉❛ ✉♠ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AB ❀ • ❈♦♥str✉❛ ❛❣♦r❛ ♦s s❡♠✐❝ír❝✉❧♦s AC ❡ BC ❞❡ ♠♦❞♦ q✉❡ ✜q✉❡♠ t♦❞♦s ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ r✱ ♥♦ q✉❛❧ ❡stá ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AB ✳ ❙♦❧✉çã♦ ✿ ✺✼ ✺✽ ❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ❋✐❣✉r❛ ✹✳✶✿ ❱✐s✉❛❧✐③❛çã♦ ❞❛ ♣r✐♠❡✐r❛ ❝♦♥str✉çã♦✳ ❊①❡♠♣❧♦ ✹✳✷✳ ❊♠ r❡❧❛çã♦ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ✶✳ ❈❛❧❝✉❧❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦s s❡♠✐❝ír❝✉❧♦s AC ✱ AB ❡ BC ✳ ❱❡r✐✜q✉❡ ❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❡①t❡r✐♦r ❡ ✐♥t❡r✐♦r✳ ❖ q✉❡ ✈♦❝ê ♦❜s❡r✈❛❄ ✷✳ ❈❛❧❝✉❧❡ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❛♦ s❡♠✐❝ír❝✉❧♦ AC ❡ ❡①t❡r♥❛ ❛♦s s❡♠✐❝ír❝✉❧♦s AB ❡ BC ✳ ❙♦❧✉çã♦ ✿ ✶✳ ❙❡❥❛ α ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AC = 1✱ β ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ 2 s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AB = ❡ γ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ 1 BC = ✳ 3 3 ▲♦❣♦✱ t❡♠♦s q✉❡✿ ❊♥tã♦✿ α = 2·π· β = 2·π· γ = 2·π· 2 2 2 1 2 =π 2 2 6 =π 3 1 6 =π 6 ✹✳✶✳ ✺✾ ❊❳❊▼P▲❖❙ α = β + γ. ✷✳ ❆ ár❡❛ ❞❛ r❡❣✐ã♦ ♣❡❞✐❞❛ s❡rá ❛ ár❡❛ ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AC ♠❡♥♦s ❛ s♦♠❛ ❞❛s ár❡❛s ❞♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AB ❝♦♠ ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ BC ✱ ♦✉ s❡❥❛✿ A = A1 − (A2 + A3 ) 2 π · 21 π = A1 = 2 8  1 2 π· 3 π = A2 = 2 18  1 2 π· 6 π = A3 = 2 72 i π π hπ − + A = 8 18 72 π A = . 18 ❊①❡♠♣❧♦ ✹✳✸✳ ❈♦♥str✉çã♦ ✷✳ • ❈♦♥str✉❛ ✉♠❛ r❡t❛ r❀ • ▼❛rq✉❡ ♦s r❡s♣❡❝t✐✈♦s ♣♦♥t♦s A✱ B ❡ C s♦❜r❡ r✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ ♣♦♥t♦ B ✜q✉❡ 1 2 ❡♥tr❡ ♦s ♣♦♥t♦s A ❡ C ❡ q✉❡ AC = 1✱ AB = ❡ AC = ❀ 3 3 • ❈♦♥str✉❛ ✉♠ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AB ❀ • ❈♦♥str✉❛ ❛❣♦r❛ ♦s s❡♠✐❝ír❝✉❧♦s AC ❡ BC ❞❡ ♠♦❞♦ q✉❡ ✜q✉❡♠ t♦❞♦s ❞♦ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛ r✱ ♥♦ q✉❛❧ ❡stá ♦ s❡♠✐❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ AB ✳ • P❡❧♦ ♣♦♥t♦ B ❝♦♥str✉❛ ✉♠❛ r❡t❛ t ♣❡r♣❡♥❞✐❝✉❧❛r ❛ r❡t❛ r❀ • ▼❛rq✉❡ ♦ ♣♦♥t♦ D ♥❛ ✐♥t❡rs❡❝çã♦ ❞❛ r❡t❛ t ❝♦♠ ♦ s❡♠✐❝ír❝✉❧♦ AB ❀ • ❈♦♥str✉❛ ✉♠ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ BD✳ ❙♦❧✉çã♦ ✿ ❊①❡♠♣❧♦ ✹✳✹✳ ❊♠ r❡❧❛çã♦ ❛♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ✶✳ ◗✉❛♥t♦ ♠❡❞❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦ BD❄ ✻✵ ❈❆P❮❚❯▲❖ ✹✳ ❆P▲■❈❆➬Õ❊❙ ❋✐❣✉r❛ ✹✳✷✿ ❱✐s✉❛❧✐③❛çã♦ ❞❛ s❡❣✉♥❞❛ ❝♦♥str✉çã♦✳ ✷✳ q✉❛❧ é ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ BD❄ ✸✳ ❖ q✉❡ s❡ ♣♦❞❡ ❝♦♥❝❧✉✐r ❡♠ r❡❧❛çã♦ ❛ ár❡❛ ❞❡ss❡ ❝ír❝✉❧♦ ❡ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ♦❜t✐❞❛ ♥♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r❄ ❙♦❧✉çã♦✿ ✶✳ ❙❛❜❡♠♦s q✉❡✿ AD2 + CD2 = 1  2 2 2 AD = + BD2 3  2 1 2 CD = + BD2 3 ✭✹✳✶✮ ✭✹✳✷✮ ✭✹✳✸✮ ❙✉❜st✐t✉✐♥❞♦✲s❡ ✭✹✳✷✮ ❡ ✭✹✳✸✮ ❡♠ ✭✹✳✶✮✱ t❡♠♦s✿  2  2 2 1 2 + BD + + BD2 = 1 3 3 4 1 = 1 2BD2 + + 9 9 4 2BD2 = 9 2 BD2 = 9√ BD = 2 . 3 ✹✳✶✳ ✻✶ ❊❳❊▼P▲❖❙ ✷✳ ❆ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ ❞✐â♠❡tr♦ BD s❡rá ❞❛❞❛ ♣♦r✿ A = π·  A = π· A = 2 BD 2 √ !2 2 6 π 18 ✸✳ P♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❛s ár❡❛s sã♦ ✐❣✉❛✐s✳ ❊①❡♠♣❧♦ ✹✳✺✳ ❙❡❥❛♠ ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ ❛ r❡t❛ r✳ r ✉♠❛ r❡t❛ ❡ ❞♦✐s ♣♦♥t♦s A ❡ B ❛♠❜♦s ❞❡ ✉♠ ♠❡s♠♦ ❧❛❞♦ ❞❛ r❡t❛✱ ❈♦♥str✉❛ ✉♠ ❝ír❝✉❧♦ ♣❛ss❛♥❞♦ ♣❡❧♦s ♣♦♥t♦s A❡B q✉❡ s❡❥❛ t❛♥❣❡♥t❡ r✳ ❙✉♣♦♥❞♦ ♦ ♣r♦❜❧❡♠❛ r❡s♦❧✈✐❞♦✳ ❋✐❣✉r❛ ✹✳✸✿ ❱✐s✉❛❧✐③❛çã♦ ❞❡ ❝♦♠♦ ✜❝❛r✐❛ ❞❡♣♦✐s ❞❡ r❡s♦❧✈✐❞♦✳ ❊s❝♦❧❤❡♠♦s ♦ ♣♦♥t♦ A = O ♣❛r❛ s❡r ♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✱ ❞❡ ♠♦❞♦ q✉❡ t❡r❡♠♦s ✉♠ ❝ír❝✉❧♦ C ❞❡ ❝❡♥tr♦ A✳ ❙❛❜❡♠♦s q✉❡ ❛ ✐♥✈❡rsã♦ ❞❡ ✉♠❛ r❡t❛ r q✉❡ ♥ã♦ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦ é ✉♠ ❝ír❝✉❧♦ r′ q✉❡ ♣❛ss❛ ♣❡❧♦ ❝❡♥tr♦ ❞❡ ✐♥✈❡rsã♦✳ ❆❣♦r❛ ✐♥✈❡rt❡♠♦s ♦ ♣♦♥t♦ B ❡♠ r❡❧❛çã♦ ❛♦ ❝ír❝✉❧♦ C ♣❛r❛ ♦❜t❡r ♦ ♣♦♥t♦ B ′ ✳ P❡❧♦ ♣♦♥t♦ B ′ tr❛ç❛♠♦s ❛s r❡t❛s t ❡ s t❛♥❣❡♥t❡s ❛♦ ❝ír❝✉❧♦ r′ ✱ ♣♦✐s ❝♦♠♦ ❛ s♦❧✉çã♦ ♣r♦❝✉r❛❞❛ ❞❡✈❡ ♣❛ss❛r ♣♦r B ❡ ❞❡✈❡ t❛♥❣❡♥❝✐❛r r ❡♥tã♦ ❛s r❡t❛s q✉❡ ♣❛ss❛♠ ♣♦r B ′ ❞❡✈❡♠ t❛♥❣❡♥❝✐❛r r′ ✳ ❆♦ ✐♥✈❡rt❡r♠♦s ❛ r❡t❛s s ❡ t ❡♠ r❡❧❛çã♦ ❛ C t❡r❡♠♦s ♦s ❝ír❝✉❧♦s s′ ❡ t′ ♣❛ss❛♥❞♦ ♣❡❧♦ ❝❡♥tr♦ A = O ❞❡ ✐♥✈❡rsã♦✳ ❆s s♦❧✉çõ❡s sã♦ s′ ❡ t′ ✳ ✻✷ ❈❆P❮❚❯▲❖ ✹✳ ❋✐❣✉r❛ ✹✳✹✿ ■♥✈❡rt❡♥❞♦ r ❡♠ r❡❧❛çã♦ ❛ ❋✐❣✉r❛ ✹✳✺✿ ❙♦❧✉çã♦✳ C✳ ❆P▲■❈❆➬Õ❊❙ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ❬✶❪ ❇❆◆❑❖❋❋✱ ▲❡♦♥✳ ❆r❡ t❤❡ t✇✐♥ ❝✐r❝❧❡s ♦❢ ❆rq✉✐♠❡❞❡s r❡❛❧❧② t✇✐♥s❄✱ ▼❛t❡♠❛t✐❝s ▼❛✲ ❣❛③✐♥❡✳ ❱♦❧✳ ✹✼✳ ❬✷❪ ❇❖❆❙✱ ❍❛r♦❧❞ P✳ ❘❡✢❡❝t✐♦♥s ♦♥ t❤❡ ❆r❜❡❧♦s✱ ❚❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ▼❛♥t❤❧② ✲ ❱♦❧✳ ✶✶✸✱ ✷✵✵✻✳ P✳ ✷✸✻ ✲ ✷✹✾ ❉✐s♣♦♥í✈❡❧ ❡♠ ✿❁❤tt♣✿✴✴✇✇✇✳♠❛t❤✳t❛♠✉✳❡❞✉✴ ❤❛r♦❧❞✳❜♦❛s✴♣r❡♣r✐♥ts✴❛r❜❡❧♦s✳♣❞❢❃✳ ❆❝❡ss♦ ❡♠ ✸✵ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✹✳ ❬✸❪ ❇❖❨❊❘✱ ❈❛r❧ ❇❡♥❥❛♠✐♥✳ ü❝❤❡r✱ P❛✉❧♦✿ ❊❞❣❛r ❇❧ ❬✹❪ ▼❖❘❚■▼❊❘✱ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ❚r❛❞✳ ❊❧③❛ ❋✳ ●♦♠✐❞❡✳ ✷ ❛ ✳ ✳ ❡❞✳ ❙ã♦ ✷✵✵✶✳ ❇r✐❛♥✳ ❚❤❡ ●❡♦♠❡t② ♦❢ ❚❤❡ ❆r❜❡❧♦s✳ ❁❤tt♣✿✴✴s❝❤✉❧❡✳❜❛②❡r♥♣♦rt✳❝♦♠✴❛r❜❡❧♦s✴❛r❜❡❧♦s❴✵✻✳♣❞❢❃✳ ❉✐s♣♦♥í✈❡❧ ❆❝❡ss♦ ❡♠✿ ❡♠✿ ✸✵ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✹✳ ❬✺❪ ▼❯◆❆❘❊❚❚❖✱ ❆♥❛ ❈✳ ❈♦rrê❛❀ ❙■◗❯❊■❘❆✱ P❛✉❧♦ ❍❡♥r✐q✉❡✳ çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❆♣♦❧ô♥✐♦ ✉t✐❧✐③❛♥❞♦ ❣❡♦♠❡tr✐❛ ✐♥✈❡rs✐✈❛✳ ❙♦❧✉✲ ❉✐s♣✐♥í✈❡❧ ❡♠✿❁❤tt♣✿✴✴✇✇✇✳❞❡❣r❛❢✳✉❢♣r✳❜r✴❞♦❝❡♥t❡s✴♣❛✉❧♦✴♣✉❜❧✐❝❛❝♦❡s❴❛rq✉✐✈♦s✴❊P■❙❚✷✵✳♣❞❢❃ ❬✻❪ ❙❍■◆❊✱ ❈❛r❧♦s ❨✉③♦✳ ❍♦♠♦t❡t✐❛s✱ ❈♦♠♣♦s✐çã♦ ❞❡ ❍♦♠♦t❡t✐❛s ❡ ♦ Pr♦❜❧❡♠❛ ✻ ❞❛ ■▼❖ ✷✵✵✽✳ ❘❡✈✐st❛ ❊✉r❡❦❛✳ n◦ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ✷✾✳ ❁❤tt♣✿✴✴✇✇✇✳♦❜♠✳♦r❣✳❜r✴♦♣❡♥❝♠s✴r❡✈✐st❛❴❡✉r❡❦❛✴❃✳ ❆❝❡ss♦ ❡♠✿ ✸✵ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✹✳ ✻✸
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